Probability, Statistics and Analysis 9780521285902, 0521285909

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oCe}ami tcidals) gershaters] mexelor(syay ture Note Series |

Probability, Statistics and: : LE es fae

a

nee

ss a

erst

Edited by

JECKINGMANand

Ue

G.E.H. REUTER

CAMBRIDGE

UNIVERSITY

PRESS

LONDON

MATHEMATICAL

Managing

Editor:

Mathematical

SOCIETY

Professor

Institute,

LECTURE I.M.

24-29

NOTE

SERIES

James,

St Giles,Oxford

General cohomology theory and K-theory, P.HILTON Algebraic topology, J.F.ADAMS Commutative algebra, J.T.KNIGHT Integration and harmonic analysis on compact groups, R.E.EDWARDS Elliptic functions and elliptic curves, P.DU VAL - Numerical ranges II, F.F.BONSALL & J.DUNCAN + New developments in topology, G.SEGAL (ed.) Symposium on complex analysis, Canterbury, 1973, J.CLUNIE & W.K.HAYMAN

(eds.)

Combinatorics: 1973,

-

-

-

-

.

¢

Proceedings

T.P.McDONOUGH

of

the

& V.C.MAVRON

British

Combinatorial

An introduction to topological groups, P.J.HIGGINS Topics in finite groups, T.M.GAGEN Differential germs and catastrophes, Th.BROCKER & L.LANDER A geometric approach to homology theory, S.BUONCRISTIANO, C.P. BOURKE & B.J.SANDERSON Sheaf theory, B.R.TENNISON Automatic continuity of linear operators, A.M.SINCLAIR Parallelisms of complete designs, P.J.CAMERON The topology of Stiefel manifolds, I.M. JAMES Lie groups and compact groups, J.F.PRICE Transformation groups: Proceedings of the conference in the University of Newcastle-upon-Tyne, August 1976, C.KOSNIOWSKI Skew field constructions, P.M.COHN Brownian motion, Hardy spaces and bounded mean oscillations, K.E. PETERSEN Pontryagin duality and the structure of locally compact Abelian groups,

S.A.MORRIS

Interaction models, N.L.BIGGS Continuous crossed products and type III von Neumann A.VAN DAELE Uniform algebras and Jensen measures, T.W.GAMELIN

. Permutation

groups

combinatorial

structures,

N.L.BIGGS

M.F. ATIYAH B.SIMON

et

group

and

algebras,

Representation theory of Lie groups, Trace ideals and their applications,

. . Homological

theory,

C.T.C.WALL

(ed.-)

Introduction to Hp spaces, P.J.KOOSIS Theory and applications of Hopf bifurcation, N.D.KAZARINOFF

Topics - Graphs,

in

& Y-H.WAN

theory

the

Z/2-homotopy

theory,

Recursion theory: its & S.S.WAINER (eds.) p-adic

analysis:

Coding

the

a

Universe,

Low-dimensional

of group

designs,

and

codes

presentations,

short

course

A.BELLER,

topology,

on

and

recent

R. JENSEN

R-BROWN

G.W.BRUMFIEL

B.D.HASSARD,

JOHNSON

D.L.

LINT

J.H.VAN

&

P.J.CAMERON

M.C.CRABB generalisations

& A.T.WHITE

al.

. Partially ordered rings and semi-algebraic geometry, Surveys in combinatorics, B.BOLLOBAS (ed.-) . Affine sets and affine groups, D.G.NORTHCOTT

46. 47. 48.

Conference

(eds.)

applications, work,

N.KOBLITZ

& P.WELCH

& T.L.THICKSTUN

(eds.)

F.R.DRAKE

49.

Finite geometries and & D.R.HUGHES (eds.)

50. 51. 52.

Commutator calculus and groups of homotopy Synthetic differential geometry, A.KOCK Combinatorics, H.N.V.TEMPERLEY (ed.)

53. 54. 55.

Singularity theory, V.1I.ARNOLD Markov processes and related problems of analysis, E.B.DYNKIN Ordered permutation groups, A.M.W.GLASS Journées arithmétiques 1980, J.V.ARMITAGE (ed.) Techniques of geometric topology, R.A.FENN Singularities of smooth functions and maps, J.MARTINET Applicable differential geometry, F.A.E.PIRANE & M.CRAMPIN Integrable systems, S.P.NOVIKOV et al.

56. 57. 58. 59. 60. 61.

The

core

model,

P.CAMERON,

J.W.P.HIRSCHFELD classes,

H.J.BAUES

A.DODD

62.

Economics

63. 64.

Continuous semigroups in Banach algebras, A.M.SINCLAIR Basic concepts of enriched category theory, G.M.KELLY

65. 66. 67. 68.

Several complex variables and complex manifolds I, M.J.FIELD Several complex variables and complex manifolds II, M.J.FIELD Classification problems in ergodic theory, W.PARRY & S.TUNCEL Complex algebraic surfaces, A.BEAUVILLE

69. 70.

Representation theory, I.M.GELFAND et. al. Stochastic differential equations on manifolds,

71.

Groups

72.

Commutative

73.

Riemann

-

for

designs,

St

mathematicians,

Andrews

1981,

algebra:

surfaces:

C.M.CAMPBELL

Durham

a view

J.W.S.CASSELS

1981,

toward

&

R.Y.SHARP several

K.D.ELWORTHY

E.F.ROBERTSON

(eds.)

(ed.)

complex

variables,

A.T.HUCKLEBERRY

74. 75.

Symmetric designs: an algebraic approach, E.S.LANDER New geometric splittings of classical knots (algebraic L.SIEBENMANN

knots),

& F.BONAHON

76. 77. 78. 79. 80.

Linear differential operators, H.0O.CORDES Isolated singular points on complete intersections, E.J.N.LOOIJENGA A primer on Riemann surfaces, A.F.BEARDON Probability, statistics and analysis, J.F.C.KINGMAN & G.E.H.REUTER (eds. ) Introduction to the representation theory of compact and locally

81.

compact groups, A.ROBERT Skew fields, P.K.DRAXL

London

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Probability,

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79

Statistics

Analysis

Edited

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J.F.C.

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Professor

of

Mathematics,

University

of

Oxford

and

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REUTER

Professor of Mathematics, Science and Technology

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PRESS

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Cataloguing

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iG VGhoxemeil, mioltses III. Kendall, David

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O 521

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82-19731

Data

Probability, statistics and analysis. - (London Mathematical Society lecture note series, ISSN 0076-0552; 79) 1.

of

CB2

1983

1983

Printed

University Cambridge

lectures

These

papers

are

D.G.

Kendall,

the

University

We

hope

as

a

Mol olSo

and

that

sign

the

of

dedicated

F.R.S.,

of

he

our

KGheeeils

authors

to

Professor

Cambridge)

will

enjoy

admiration

Cipalels

of

David

for

and

papers

(Professor

Mathematical

his

reading

ieibhe cha

these

of

Pendalt

Statistics

sixty-fifth

them,

and

friendship.

will

in

birthday. accept

them

Digitized by the Internet Archive in 2021 with funding from Kahle/Austin Foundation

https://archive.org/details/probabilitystatiO000Ounse_k7j4

CONTENTS

The

asymptotic D.

On

doubly

shape

population

for

occupation

BINGHAM

and

J.

Martin boundary processes CRANSTON,

E.B.

a

of

Invariant PP.

24

times

46

two

dimensional

Ornstein-Uhlenbeck

OREY,

U.

ROSLER

for

a

symmetric

Markov

9)

of

Kabanov,

EAGLESON,

Liptser

R.F.

and

9)

Sirjaev

GUNDY

Nak

Ball

HAMMERSLEY

measures

and

the

143

q-matrix

KELLY

appearance of a multivariate exponential distribution in sojourn times for birth-death and diffusion processes Dele

Three

problems

in

discrete

180

theory

capacity

of

an

Ley

ellipsoid

MORAN

Stationary one-dimensional Markov continuous state space F.

Markov

KINGMAN

electrostatic P.A.P.

161

KENT,

unsolved J.F.C.

The

processes

63

S.

Commemoration

J.M.

The

system

DYNKIN

theorem G.K.

Oxford

a particle

HAWKES

Green's and Dirichlet spaces transition function On

of

PITMAN

theorems

N.H.

M.

and

J.

BARTLETT

limit

The

and

stochastic

M.S.

On

speed

ALDOUS

PAPANGELOU

random

fields

with

a

ISI)

\iginiatial,

A uniform central limit theorem processes indexed by sets Re

ZO

randomness

properties of a test for kernel density estimates B.W.

Criteria

for

R.L.

241

multimodality

based

on

248

SILVERMAN

rates

application

to

of

convergence

queueing

and

of

Markov

storage

chains,

theory

with

260

TWEEDIE

Competition P.

partial-sum

PYKS

Multidimensional B.D. RIPLEY

Some

for

and

bottle-necks

PENA

WHITTLE

Contributors

285

THE

ASYMPTOTIC

David

of

a

system

IP

elneroductLen study

of

and

original

an

The

connected.

(A box

occupied

ied

box

from It

and

the

clear

speed Sy ~

of

k-ball s ireelf e/k

THEOREM

(1980)

pure

each

box the

set

is

right

of

to

infinity.

to

infinity

1.1

As

k

increases

result

arguments

Weiner

was

(1980).

(2.3)

to

the

growth

process

(defined

the

the

the

*

motion

integers.

and

and

This

our

is

Our

the

results,

the

no

from

k balls

empty

the

easy

to

are

boxes box

uniformly

mutually

almost

as

occup-

at

random

independent. and

see

that

that

certain

result

is

between

left-most

connected,

main

sites

occupied

chosen

is

an

at

by

of

Tovey

proof with

for

the each

average

is

that

increases

to

(private

have

communication),

been

(Sections

a certain,

e.

given

2-4) more

is

by to

easily

Keller use analysed,

(3.3)).

k we

can

equilibrium

Research supported in part by MCS80-02698 and MCS78-25301.

ks)

conjecture

process

fixed the

of

system

of

being

It

at

infinity,

method

k-ball

describing

there

remains

below.

conjectured

Our

compare

SYSTEM

precise:

supporting

to

..)

by

more

set

taken

choices

off

be

by

»

Initially

a ball

boxes

off

Secondly,-for

*

but

a ball

drifts

formally

k >

balls,

the

balls,

drifts

To

as

that

balls

~.

to

follows.

motion defined

as

particle

many

occupied

k >

and

as

successive

of

PARTICLE

study.

a way

the

A

labelled

the

move,

to

as

informal

coupling

one

sites

of

referred

such

contain At

the

This

and

in

OF

asymptotics

at

the be

described

k balls,

that

collection

k the

be

may

for

SHAPE

Pitman

the

located

will

boxes

boxes.)

placed

among

is

may

Jim

paper

motivation

amongst

AND

description

particles

motion

two

this

informal

The

distributed

in

k particles

gives

boxes.

and

We

section the

Aldous

SPEED

.

National

define

(Section

proportions

.

Science

5)

of

a random

balls

:

Foundation

in

grants

the

vector (Oth,

lst,

2nd,...)

most

box

has

box

from

the

just

been

cleared.

(1_,1.,.,+--) Ome laae

i

IS

Gonverges

IS

(DorPy Pore s+) large

k the

of

process

problems

concerning

positive

integers.

The

of

and

on

with

of

the

the

each

connected of for

which

it

must

In

other

maximum H

by

an

on

a

the

maximum is

i,

vertices

local-global

and

There

is

the

with

v

at

Now

SUGihie

i

are

is

for

almost related

to

measures

on

Tovey

(1980)

of

functions

distinct

the

f

real

values,

any

such

v

is

a

is

this

number

that

local

f£

maximum oy

which

case

average"? to

some

induces

el(you) 2 me amrton

the

in

"on

required to

> #i(sq)) are

neighbor

maximum,

steps

function

they

locate

the

algorithm

of

f(i)

if

to to

according

(cvie)ame> 1

maximum.

neighbors

algorithm

random

teliercammt:

global

a vertex

good

expected

picked

the

obvious

from

How

functions?

VEEERCES

work

Consider

cube,

a vertex

unless

maximum.

f

an

move

largest;

is

except

maximum

function:

what

evolves

probability

algorithms.

d-dimensional

that

property:

local

of

is

ORE

the

;

conjecture

in

left-

k++»,

imply

process

this

process

improvement of

how

as

the

sequence

would

k-ball

came

of

function

VyirVar Varese

the

k-ball

global

words, of

in

show

:

of

edge.)

the

This

transformation

j

f(v')

be

es

when

that

certain

certain

a local

a local-global

a

:

Pea

times

(5.9)

to

a

no

neighbor

A

with

at

6 we

local-global

(Here for

conjecture

distribution

proportions

local

has

We

box

Section

vertices

£

occupied

In

origin

abstractions

defined

in

constants

deterministically.

on

leftmost

locate

the

distribution

an

3 ayn) ae

ordering cite

Tekalel!

35

TselLocall—Globaliairt

uA

Thus

is

a

neighbor

a distribution

Vi VorVarees

steps

wu

on

of

at

least

local-global

OLRVEGELCeSeSatistying

required

by

the

algorithm

one

{Vy rere r¥

functions

(122

started

of

induces

net

at

N,

vertex

V.,

the

(N, )

number

of

satisfies

a

Ae

steps the

i]

required

from

the

"worst"

initial

dees gy

2

(Cre)

a random

denote

a

is

n-1

so

the

that

vertex.

ordering

number

max

i

of

N.

Plainly

recurrence

AL ae Nu

where

j < i is

bs is

a neighbor

the of

(GS) least vy,

i

integer

for

which

But

we

can

think

corresponds) the

number

evolves

N.

by

of

the ball

ing

And

balls.

vertex,

is

To

rightmost

the

of

spirit

2.

k

where

some

this

of

of

of

steps

rightmost

we

need new

an

the

distribution

the

the

be

used

rightmost

k-ball

wy

But

to

is

show

process:

see

existinitial

balls

speed

of

This

have

the

process in

some

arguments

that

for

box

Tovey

box

local-global

coupling

occupied

the

chosen

on

to

the

worst

oe

the

j

of

from the

process.

because

increases.

3 can

of

in boxes"

way

after

on

i

process

right

from

box

bound

the

the

random

occupied

stuyudy--

balls

to

ball

corresponds

says

box

required

upper

"balls

we

Section

speed

of

in

certain the

new

process

(1980).

Lx

x

Gerarel

measure

is

#{3:

number

k-tuple

of

of

balls

nonnegative

in

boxes

numbered

integers

Rx

ES *“

i

B

g

i]

3©*

h-

3

connected

balls;

box

of

number

regard

without

are

balls ball

assumed

number

to labelling

i=O7 17. -)) Gefined

in

of

box

i

configuration

tad

i}

min{i:

N.x SSO},

*

ii}

max{i:

N,x Ol,

if

the

j.

is

to

be

The

recorded

by

L=O7L yrs sen

balls

of

of

the

(NX,

a

end

right

is

boxes

Nx =

number

the

zs

among

balls

counting

and

a

finite

number

total

the

is

of

is

by

a

(X) Xone +27)

N.x =

N.x gigs

of

described

Un acophsh

distribution

the

be

=

#x

=

ilevoySibilecl

and

in

configuration

will

x

end

chosen

where

i

Preliminaries

0,1,2,.-.-

So

j

is

boxes,

ball (1.3)

the

number

in

Then in

the

speed

balls

containing

placed

involving

of

of

box

algorithm.

process

number

the

the

The

by

of

way

those yw

the

this,

box

the

distributions

than

of

the

being

the

k-ball

random and

in

where

position

a process

and

ball

estimate

functions,

less

j,

the

complicated

is

steps

occupied from

as

Me,

Sees Noy

the

used.

differs

(N, )

Gary"

containing

been

of

tol vertex

configuration

for

x

are

defined

x.

by

The

left

N.x

The

set

Wen

pute

a

by

of

53 0) Nmohe

connected

Coa=

UL.Cy +

Bs oe Bs

configurations

FOr

xe

Che

il

except

J

= x

where

4 =

box

Lx.

from

box

4 (x) That

k

balls

will

be

define

denoted

a new

Cyr

and

configuration

and

for

all

k Markov

matrix

the

to

number

say,

x

replacing

if

j =

)

LE

7.=

3

of

the

lowest

is

obtained

in

the

.

td

is is

Lx

xt ence

it

i-=1,...,k.

chain

PB, (xy)

with

; lim

S. =

m

The

countable

defined

(X(m),

limits

ale LX(m)

tion

X(0).

To

see

=

L

where

for

L Nx

L (N5X,

=

a

is

surely

why,

(N-X(m),

lim

for

x

m

=

of

ball

process

and

the

one

is

step

left

ball

end 4

i.

Clearly

the

discrete

transition

whe

12x (m)

F

(Ain 33)

:

discrete

k

the

the ¥

O71.)

given

right

C.

in

removing

do the

not left

k-ball

process.

depend

on

the

counting

We

assert

initial

that

configura-

process

m = O,1,..-)

(detaneds

A Indeed,

a

and

consider

configuration

i=

the

ball

by

k-ball

space

m7

almost

to

x

t= ocak.

m=O0,1,...)

exist

box

numbered

from

discrete

state

m-co

where

(2)

by

Dime ) = 17k)

the

of

i=1,...,k, ‘

tee

time

Wrens

left

count

of

x

is

the

vector

by

20m

left

counting

(2.4) process

is

a

Markov

chain

whose

finite

state and

ae ns =k

This

space

motion

> (0) seehe

is

easily

k-ball

process,

tical

to

a clearance

the

some

of

we

(2.3),

=

of

the

event

move

m.

Thus

=

other

use,

but

in

ees

since

RX(m))

(ii) =

N(1)

example,

N(m)

implies

and

that

when

clearance variables,

from

tT)

let

give

N(O)

here

Recall

is

= inf{m:

in

(m=)

the

ei

shall

we

of

a

seaden—

refer

(2.5)

S,i

one,

for

an

which

will

irreducible

A, \

Section

Lf

|v(O)

ona

set

be

the

Markov as

of

7 describes basis

chain

given

states

the A,

then

Y ea} m

1/A(A),

(1971),

No=l),

the

to

the

left

and

Y(T,)

Section

is

2.5.

Ala: Applying

definition

N, (1)

has distribution

above

is

time

identical

of

only

conditioning

expectation

(n:

Pex

which

,

speed

distribution

by

return

the

that

the distribution of

equilibrium (i)

us

5.

Freedman

A =

there

definition

EXGn)

(nox (m-1)=1)

for

A

the

so

Ss kK:

equilibrium

has

yY(m)

Now

aperiodic,

variable

expressions

T

such

obviously

obtained

the

0)

O~

LX(m)

Section

wlth Ala

(i)

for

say.

random

and

lim m TESA c Nex (4)=1}

distribution

See,

irreducible x

the

i2

bh & Gl

moo

not

WL),

to be

21

(n,, i

with

E (6) sxene

tel

n= ~

: -1 lim m ~LX(m)

developments

VO)

at

several

do

m

vectors

mow

©

are

seen

counting

d21

al SS Cl

indicator

justifies

There

of

distribution,

for

to

set

exists

ig,

equilibrium

This

the

there

unique

as

is

eis

of a

A,|(n: n)=1), counting

conjunction

with

process. (2.5)

the be

this

k-ball the

After the

to

process

distribution

and call

yields

fact

v, the

a change formula

of

of

Wich

where

the

tion

Vee

the

left

For

an

4a

right

We

Ip

side

denotes

record

also

counting

process

if

has

N(O)

arbitrary

(2.6)

for

initial

limiting

is

of

the

oie

tion

Ve

of

the

3.

Speed

of

k,

is

To

t > O

with

and

is

later

is

te

i

in

entries

€20)

not

of

has

(ii)

distribu-

above.

For

N(1),...),

sure

Viet then

the

limit

comparison the

state

nace

by

Wy

so

does

N(N, (0) )

distribution

of

N(M_)

as

n+

similar

are

=

is

;

Vy

3

Cam

(ART)

Still

as

n-+>o,

o

of

the

empirical

Sy

for

and

C.

;

apls

be

where

Mo

distribu-

values

the

Markov

parameter

rates.

of

transition

are

the

row

assumed sums

rate

to

zero.

be Put

matrices,

zero,

and

another

off-

the

way,

iff

2)

a discrete

k-ball

time

is

(Baas)

process

k-ball

Poisson process with rate ; Sil Since t M(t) > k a.s.,

continuous

different

This

3

mentioned make

process.

continuous

transition

Sao

to

speeds

space

k-ball

1

the

k-ball

descriptions

taken

Me)

of

continuous

the

explicitly

a continuous

oy: WHE Tem Yn, independent

now

specified

diagonal

the

N_

N(M)) »N(M5),---,N(M_).

countable

which

rates

that

when

clearance:

facilitate

diagonal

(XXr

No

a consequence

distribution

almost

introduce

UG ae OS

Here,

of

comparisons

we

process

the

sequence

use

distribution,

the

Further,

later

distribution

as

time

expectation

(N(O),

interpreted the

the

process

has

process

and

(M(t),

£20)

is

an

k. a

comparison speed

with

(2.3)

above

shows

lim

LX, /t =

lim

too

Notice

that

NX t!

so

(NX,

both

the

LX,

speed

t20).

Now

so

box

une

together

most

occupied

each

ball

were

and

a

one

ball

suggests

rules

described

birth

process.

where

x

i+

is

but

l,

is

defined

’

J

St

simple

case

of

processes

to

due

completeness

PROPOSITION

in

with

the

of

if

other

box

annihilation

the

call

leftas

original

each

according We

the

proceeds

with

with

in

split.

of

This

Markov

process

to

splitting

this

the the

lateral

are

i=l,..-,#x

A

(Bins)

by

are

~

birth

results

for

(1975)

branching

provide

a proof

3.4.

For

a lateral

birth

in

birth

next

process

result

is

processes

and

(1978),

but

for

Section

process

is

a

a special

Markovian

Mollison

shall

RB, /t =

The

process.

and

lateral

the

(1978),

Mollison

Kingman

-7 i

we

lim

in

the

ball

eet

contact

general

more

Se

ede

of

terminology

particularly

process

a ball

independently

right,

evolves

view

a new

4 ee

J

the

the

annihilations.

rates

rate

alee Db

In

k-ball

no

balls,

can

of

motion

remaining

to

one

of

the

measure

process

creation

simultaneous

which

Uy C.

with

box

counting

measure

view

two

ball

box

continuous

at

€ Ce

one

the

concerned, the

of

1 into

occupied

by

annihilation

"mother"

transition

i+ x"

>

are as

point

rate

a

~€, =

above Its

x

(B22)

counting

counts

appearing

the

space

the

process

this

at

leftmost

comparing

State

‘

determined

simultaneous

From

are

ball

the

by

these

with

Lx.

balls

"daughter"

whose

ks)

are

a k-ball

splitting

two

in

as

of

box

oe

determined

far

x > xXj

These

and

is

transition

balls.

RX, /t =

too

the

contact

of

sake

4.

(B_,

t20),

8 eicSc

t+

To

compare

the

progress

of

different

processes

of balls

in

boxes

we

introduce

one,

ys}

a partial

Bele

ole

j2h J SQ

8 S yy

h.

(To

abe

Strichilys i=

hti;

let

the

to

the

in

night

of

x

whose

#x

={ #7

wane

and

Were

shows

~s =< y

#x

e

State

spaces

C

Q

COnEUgunatetOns A and an M-chain

a

x

is

eight

Of

behind

label.

plus

the

each

r=

lefts)

is

number of

to

of

to

Given

according

to say,

yr

in

in

the x

this

to

see

ball

Nori

rank

tk,

Ge

sie)

thes yr,

Toos

(s7yy,

REMARK

In

A

process

is

A

3.7. (X,X)

an

M-chain

EVOMuEROn mn xa

PROPOSITION

mapping

I

Se

Bight

S ot

from

is

an

the

applications

below,

(ii)

can

then

X

eG

~

be

is

derived

hs

eeXeeimakes

continuous

ahead

of

time

T

simple

£(x,%,y)

k

the

the

is

?

is

ranked

oO have

now

1

in

denote

o,ie =

ks ie

A

X

one-to-one.

transition

rates

with

simply

by

elke

the

A

of

The

by

that

saying

X

make

A

if

A

x

a transi-

A

ss Omn ye

process

process

range

equality.

letting

k-ball

process,

abhtains

of

in

mec oo, vie

stays

k

< f,

process,

birth

applications

obtained

more

bard

Errse

igh

not A {xs 5 6

2

(35)

above

to

be

valid,

the

operator

2

—~Z do

so

that

in

equilibrium

re sr leading

to

the

3/3¢10M/9o

solution

9M/dd

or

(Go7§-1)

iL

= A(1l - = 0°4)

OR /ea

= O,

(36)

34

M

in

agreement

5

(l.-

with

=

(34).

However,

operator

6 =

2

1

(

=

the

multiplying

includes

the

additional

in place

of

-2«k/o

between

the

ambiguity original

2

the

where

==

log

nis

du,

=

(kK

E{dz, } OF

the

fal))Se) 4 = on,

this

example

equilibrium il

form,

yield

an

Stratonovich

in

the

of

to

would

the

square

operator ;

an

index

and

suppose

we

Z

:

tm

1317)

eee ambiguity

arises had

the

which

-2k/o

well-known

calculi,

Thus,

from

written

the the

form

the

stochasticized

-

P

leading

example

moments.

sections

a complete

‘

one

or

ie

+

n,)dt

2

version

is

naturally

dadZ e

2

} =o

(38)

dt.

E{dn,/n,

formula

On

-

use

a=k

= @

bk, +: Ko)r

averaging

this

terms

of

$ =

so

M(o)

the

i

(an, )°/né Sse

for

the

again

For

te

aes LS)

m(n)

consistent

demonstrate

(22).

10,

to 55°

approximation a

may

earlier

ete

by making

J(k)

and

of

equation,

we

find

dit = E{du, } = E{log[ (n +dn,) /n, 1}

natural

(ig =

We

‘ is

model

i

that

This

E{ (d2Z,)

(are

so

7 9

Ito

deterministic

Us

«(t) nine

term

conditional

du, /dt

where

+1.

so-called

in

formalism

that,

the

with

+ noF/dd

Ko)s D(6)

this

the

consistency

of

relation

second

of

our

model

example,

write

Eo)

would

solution

general

switching

logistic

(ky =

from

with

he

(16),

approach or,

ee ole3/36=87/ 967 Iie G =

= (6),

(22)

be

above.

in

for

its

od{96

in

becomes

=O ,

(39) nooM/do

+ [od(k)

- 2eJF(¢)

i]

oO

ee,

where pd(k)

J(k) -

2€

A: = Kd/dp

-

and

second

the

2 2 9/36. by

. : Multiplying

the

first

73/36

not

immediately

does

of

these

equations

enable

us

to

by

35

eliminate

F,

as

these

two

operators

r)

, a6 dow al fag

[od (kK) -2e In So 36

and

the

first

equation

(ene

whence

we

do

of

(39)

not

commute.

; (ie) 26)

=

=

However,

ile)

we

have

5

yields

JOUR) MCN

nu

E(G)

=H00y

find 2

+t

[oo (k)-2e JF (KM (>) —n E1499 /96]3M/96 +(k- 5p) Xr This

:

tion

equation that

through

is

ce

by

exact,

becomes

2e

and

agreement

with We

from

the

large,

le

rise in

to

the

writers

that

is

more

generation

necessary

to

genetics

two

alleles

Let

the

1 +

s,,

frequency

/e

our

2

=o

may

solution to

,

to

process

the

for

be

for

by

dividing

terms

of

(40)

O(l1/e),

(41)

with

calculus

time.

limiting

the

case

treated

agreement

continuous

seems,

obtain

other

assump-

above.

always

relevant

we

approximating

= 0,

'white-noise'

in

usual

neglect

adhere

Stratonovich

in

models

and

make

solutions

parameter(s), must

be

with

caution.

the is

equations

misleading;

the

diffusion

equations,

approximations for

The

for

in

in

by

processes

that

derived

involved

example,

give

discussion

for

suggestion

approximating

precisely

to

conclusion

appropriate

however,

more

though

expected

general

A further

reached

the

from

it

is

passing

to

population

problems. Thus,

example,

the

consider

deterministic

we

must fact

intrinsically

calculus

n

that

stochastic

in

2

but

extreme

3 is

now

[1 + 93/d¢]8M/36

second

and

specified

the

that

we

continue

section

previous

Ito

the

shall

difficulties

this

discrete

+ 59

superimposed

recognizing

if

assuming

J(K)M(o)

in

but

is)

consider

AL and

A. at

where

Ss)

Ewens,

1979),

x

of

Ay

situation discussed by Gillespie (1974) of F ; : an : ; locus, with diploid genotypes ast ALA, AJA,

a

additive,

be

'fitnesses'

the

and

S5

are

and

denoted

be

small.

By

shown

that

it

may

be

is,

per

generation,

by

standard

the

1 +

Sy:

formulae

change

Ax

ile 57'S} +

1+

(see

in

the

S,)1

for

relative

36

1 a atte x) (Sy s,)

7

For

constant

Ss)

F

However,

if

variances

assumed

+

and

Sar

Ss, and

and

s, (1 _- x) X +

S, vary

Ss)

that

and

Oy"

Sor

Ss)

So

Hi

\ 3

T5

may

#002)

1 depending

that

with

and

E{Ax|x}

© or

such

covariances

for

(remember

1+s,x

=

E{s,}

small

be

positive

the

Wye

and ‘po, 05.

2 Wy

on

sign =

E{s,}

then,

compared and/or

Cuy-ugrH9, (91-79)

of

$1 =

Uys

under

the

d with

2 Oy:

wiith conditions

etc.

negative),

(1-w

So-

we

find

09 (004-0,)1. |

(42)

and

in

these

the

(In to

a

are

the

variance

‘approximating denominator

the

conformity

-

= Gx

with

Clearly, model

for

from

2

(9, =

x

in be

Sy Ax

formula

an an

size

relation

equilibrium

mus AP 5

for

when

above

conditional

course,

population

|

Notice

the

and

S,

could

be

(33)

infinite

that

distribution

to

insert

population.

additional

N.)

with

moments

contribution that

the

deterministic are

fixed,

when

the

neglected.

there

for

x

is

now,

if

|o|

for Zeca te

2

o,) J

me eo

form

Rix) in

the

of

equation

an

+

appropriate

automatic

verified

Z

equation

hee

Aly

one the

be

te)

(9-290, 05405) i

will,

expression

variable,

ao

of

no

Phig,

the

on

have

differential in

s

(Ix)

there

depending

(42)

It may and

2

diffusion

population

approximations

s

4%

(unambiguously)

approximating finite

ap

e

var{Ax|x}

precisely

procedures checked.

are

and

(lie x),

Gillespie's what

is

findings.

most

unambiguously

subsequently

(43)

important in

the

introduced,

is

first their

to

specify

place. accuracy

If

the

process

or

approximating

may

then

always

be

37),

4.

We Consider

R=

the

AR) +

EXTINCTION

shall

now

we

O

LA

particular,

THE

BIRTH-AND-DEATH

derivation precess,

of for

PROCESS

extinction which

probabilities.

in

the

O

O

2

O

-2

3

equation

1@)

< =O)

} yp

Aeiel

(55)

((5)5)

Wg

40

that

(53)

reduces

Be

2 (3%)

Wabtskien

us

let

solution,

exact

the

sor

es

-U

A -

Ux

WW)

Eo

902 (x)

\-ux

ox

c

(57)

i

manner

orthodox

an

in

(57)

Solving

0.

x =

=

b

case

(@),,

#

a

O,

=

%

a(x-1)

:

when

Pir

i=

==

A-

X >

the

further

consider

(for

with

arises

that

difficulty

of

kind

the

illustrate

To

O.

c =

b =

case

The

(56)

Gee

wh

(Ga

yields

=

aL (x-1)

Where

or (x)it=nt

to

we

x,

=

bees

= Wisin

e

W(x) l/ox

1) =

w/a

=

=e,

=

G(x=1)

ulva

=

-

Ga.

=1

pats

: i

(58)

Integrating

§(58))

fron

@

obtain

-— bee t(x-1) :

Gai)

WiGc))

=

ewe



2

.

(1.6)

E

Hence

the

considered

Dirichlet

up

to

space

and

certain

functionals sense,

dla tO

Fukushima

Green's linear

be

m-equivalence

Gee) yy =

(Silverstein

can

space on

H:

with

o, (£)

if

the

as

inner

a

space

H

of

functions

k

E

(lwp)

the

H

be

on

product

:

call

can

realized

extended

realized «

K

as

the

corresponds

Dirichlet

space to

K 94 €

space.)

of K,

all then,

continuous ina

82

2. Operators function

2.1.A.

and by

6 ,

We

the Borel

in

2

L

denote

Zt associated

(m)

by

R

o-algebra

35

in

the

R

‘ with

open

j symmetric

a

See transition 3

positive

half-line

(0,~)

+y

Re

A is

a

(stationary)

function

P, (xB),

transition

E>

0,

Seierg13}

J p, (dy) p, (vB) = pl, (x,B) E

The

corresponding

Two space

Rr’

operators

important

ace

On

of

functuens

transition

by

the

corresponding

ye Sane)

to

the

in

the

(isi

Euclidean

corresponding tT,

to

Brownian

motion

(BM)

= Qullee “yy n/a

the

are

exp(

a

w_(%,dy) ite

Ornstein-Uhlenbeck

given

by

the

and ies

tee

ite BS | eee Dtixe y process

(OUP).

The

T £(x)

= /ff(e ee

(2.3) corresponding

formulas

D f(x) = fie + /e 2) (az) for BM

where

formula

functions

are

ee ie

operators

eee

examples

Ts

vies”

z)y(dz)

(ap for

OUP

(2.4)

Oe

83

2.2

function

p

Let

is

m

called

be

a

Nee m(dx)p, (x,B)

The Lebesgue

measure

relative

to

m(dx)

it

function

=

measure follows!

measure

relative

on

to

dx y

from

and

(2.1)

the

(Gi,fe Vn

m

= im m(dx)p,(x,A)

transition

the

o-finite

symmetric

for all

is

by

(256)

that

A,Be

symmetric

transition

given

AN Teleshalemenvesa

if

B.

relative

function

(2.2)

(2.6) to

the

is

symmetric

(2.5). for

every

£ > 0

¥

m(T, £)

(i.e.,

m

is

an

+m(f)

measure).

We

as

put

t+

(f,g)

0

Cay)

= m(fg)

and

fiell= ce... YP

Proposition

following

2.1.

a

Ge

Sie

< T(E) (x).

follows ack

andes Dei

=a

Gy

Ps PRACE e

(T) ie g)

Des RSD

in

L

2

(m)

and

have

the

[|(r ££)

5

Sais

(£, it ao

=

= (IT,

ll

is

a positive

monotone

decreasing

ic Proof.

se

act

2.2.a. |lr,ll < [lll

SEUIONELERL@VE OIF

Ehatey

qT,

properties:

Zotege

It

Operators

By

Integrating

from seam

smcne

2.2.A

nSsustrongly

continuous

EOE

Lew

Oe rao ns

satisfy

the

relations

TY

inequality,

to

m

2.2.

and

using

for

(2.7),

m-equivalence

Om

every

re le.Diy erm

we

of

t,x,

get

(T,£ (x)?

2.2.A.

functions

OlLows

from

and (2.16),

2.2 -A, B,C.

For

has

and

preserves

Ley aBiEOlLlowsi

CatlonsOr

Proposition

Schwarz

relative

that

ate

inp

the

every

limits

f e€ L

;

ian

i

2

2

(m))

(m),

as)

the

©

; function

tends

to

©

and

84

To

is

the

orthogonal

projection

on

the

subspace

Lo =

{£:T)f

=

f}

ae ©

is

the

orthogonal

projection

on

the

subspace

lk, = 2

(2-055

=

f}2

at

every

Proof. te

By

[0,~].

u

fe

has

By 2.2.8 to

Eo

7) WE

|e

Ss

=k

|

as

from

Fe ae eh

u

s

strong

oe fee.

E+

function

follows

one-sided es

s->oO,

a, a

the

E|

:

has

cheer,

one-sided

limits

equation

a

limits

©.

Si

=

(uts) /2

ft.

in

L

|) for

S—

for

2

(m)

at

S:< b.

all

each

te

Since.

(O,~).

oF

The

[0,-]

and

st

rest

of

f

tend

So

Proposition

alNVaoHOS For

and

It

the

2

||£ -£

that

2.2.D

and

consists

both

of

BM

and

constants

2.3.

OUP,

for

Proposition

Ly =L

2

(m);L_

contains

only

O

for

BM

OUP.

2.3.

Formulas

ne

Gxa(ip=ef

G2 Ledu, O

Tot (x)

AES)

define

bounded

(250)

u

(30)

=

C2 2})

self-adjoint

linear

operators

in

(mn)

with

the

following

properties: ExS

Pc

ee

ee

Peau ec

ae Gt;

(2213)

iS)

eS

SAG f =isG Af = [

Benda

s =

REOOGe BY, @2ilasi, Fubini's

theorem,

2.2.C,B

and

S

at

eae

O

(2.14)

= Re f Se Rs

TF (x) A,

we

is

get

Measurable

that

Re

v

in)

uy

|£/aull?

x.

Using

< |Je|Pe?

7

O

Hence L

2

(m).

the

integral

ae

Similar

(2.11)

arguments

converges plus

change

absolutely of

and

variables

defines in

an

integrals

element are

of

sufficient

85

Bomprove,

(2.13)

and

2.4.

is

defined

by

the

Gf

in

the

exists

in

L

lim

Do

2

fe

ae

oe Ly

and

T

f =

0.

Green's

operator

Gf

which

(m).

Af

put

formula

=

domain

(2214).

We

(225 1055)

consists

Analogously

=

lim

Af

of

we

all

£ ¢

define

‘

the

f

such

that

infinitesimal

the

dimit

(2715)

operator

-

(2256)

s>O

in

the

domain

D. Sue fede.

A

BM

and

for

OUP

Buncitrons

ib 2

= SS Mele) £or

A

is

ss jac

(256)

2

with

exis ts: ein compact

the

amere NI

= See Ga)

Laplacian

and

Yy

is

2

the

for

By passing

to

Proposition

=

GA

the

2.4.

taf,

for

ayia)

Af

limit

If

in

fe

Hence

(2.14)

Dar

(2253)

oup. we

prove

the

following

then

¢« De

an d

-GAf=f.

iicnD:

and

-AGf

(225 159))

then

—SAeGtr—sGer

s

ffeil,»

D

BM,

-(a£,£) = 5f|ve(x) |v (dx) for

me

to

(2517)

gradient.

1l

Der

belong

OUP,

ib

eS

support

iL 2

NaS)

~(AE,£) = Sf|VE(x)|“dx

Tie

Tiny.

and

eos)

where

i¢ f,limit

s

iG

A

=f

(2.20)

then Gt

€ Dy

AG, f =

tA ‘ f ;

(45 Bi)

86

sare)

Asse s

SGA

G

Propest ica tron

=

(the

derivative

,

=

6

-1

3.

A

ae

ee

a rrr i D

then

and

from

-]

:

the

relations

6

(Tf

idee

=? ee

f)

=

Dig DRAIN 5

K

and

real-valued

m($)

(ZZ)

(m)).

spaces

2

if

2

and

The

3.1.

K-function

L

follows

(Tf oeT.T,£)

ela

(2222)

=G_f s

£ =

T At = AT, £

in

This

Si

s

, x),

and

SiO

TA, (x)

xe:

=

44

aes

(*)

for

cal

ede

all

Syme esc

PrOpoOsteion Rollsec

Qh en elf

yar

Sesh

iim

eS

Taek

aun Colom,

ls: ree: B-measurable

wLUeins

int,x.

R

Sod S135

Ilo, Il decreases.

BinduaGs

oe is

3.1.D.

$ = lim $, exists t+0

3.1.E.

Formulas

continuous

in

L* (m)

and

tends

to

a limit

.

ase

Oe

=

Tf

if and only if

Ile

is bounded.

(

i

(31h)

i

(Be) tO

establish

a

1-1

correspondence

All

these

3.2.

that

between

properties

Green's

space

(two

functions

2

i io Il¢, [Fat

Or

De =O

16 we

u

and

x

(3s2)

s>+O ,u>2

where

w

Siw ’

corresponds. Proof.

+

Rex

bh

wand

2

La)

to

we

We

consider

observe

Ehaa

2

SWire

Lew)

We

a

the

DimeEOmmurlcam

product

2

Ke

Liv)

have

Suppose)

eel,

L 3.1.B. Powe

>, =

n o. a

Hence Ss)

CCS

ve M-a.e.,

statement

of

£0r

T° O
tllop- oll

dt

K

for

m(dx)

on

converge every

= =

aes

eNO mE SUCh

mena

Te, — aholel

Tobe

is

closed

an

The

second

statement

statement.

=

that

ca k-runCELOn

Hence

3.2

vw(dt,dx)

in

t

by

(v)

ENA

mEOrCONSt mice

every

prove

Indeed

ej

Nols

Proposition Let

s :

every

: aig

De

(ie) ne

measure

If

ie nO

6 €

1)

K,

and

is

then

M=2 sc

the

first

obvious. es €

t

for

mol; Soden C an d is

cn

integrable.

For

every

expression

tends

Wy

and

one

Iv" emo lk mee By

3.1.B,C

@

aS

8

the

and

22 ©,

,

Nese

dominated

2

ure tellge

convergence

theorem,

function

A real-valued

Ss
0},

lt|, s2

£ =O

and

all we

on

Es:

such

that

sets

B_

can

I allfeels

modify

for £

in

91 Let

B=

{f,> 0}

wie ween Hence

Ree

B

is

element

with

satisfies

closed

under

E, = {p > 0}

Poe

ag

|/£ ||, < 1

have

£f =Oma.e.

implies

and

Lemma

j B=

(fee

Ell, aril 4.1,

Here

By we

10)

||£]| al,

B

contains

p 2 0,

|lpl|

an

< .

(y)

=

0

E

,

ate

Cc

we

cla

58 53 Op

||£|| < G

©

have

on

EA

(i).

We

Eg

By

mod m.

on

IT .0 ll, < lel,

and

neh

Simsae-

and

_

Conservative BM

in

R”

is

then

Tf

=O

Enda

WG

Kc

a £ yim (Ee)

ssc:

conservative

for

Zc

dissipative,

section

8 E

=

are

ae.

conservative

He:

this

conditions

eWLIE

implies is

spaces

5.1.

x,y.

|G,

tml

CHSeijoeelye

Be Seihe

all

following

fouEeviery

(ea)

m2

almost

The

m-a.e.

for

K. we

consider

and

male

only

=

0,

dissipative

then

for

semi-groups

every

p

=O

92

m°|f£| < c, (ae £) where

lee Aol,

and

m (dx)

Proofs 1.

= p(x)m(dx)

|2) act

and |/T.|£| - |£II|°> 0.

ree ae

Se Indeed,

then

by

loye

-(AF°,F°) < -2(Af,£).

(2.22)

-sAf°

(sar ;t [e|yo=. ({e=0r,|£|

=

-sA_|£|

Tie|Pauesey s

f eS. s

s+u

O

3.

Ny

the

Schwarz

< eine

By iig

|£| -

Foe

|£|

in’

s

< (GAF’,AF

W/Z tell /

am)"

2, fides

inequality

(F°,9) = -(GAF,p)

Fatou's

=

T.|f-

Hence

= (re £8) = ~ (seu) (ae. e)

Wsi sake),

aterFoh 2 285

as”

and

oar ths

de

Ghd

s

SPR A/

) “Alo ll = (F ,-AF )

‘oll

5

(5.58)

sis,

Yandi(S.1) stollows®trom

1(5.. 3)

there

We

exist

say

that

sequences

a

function

ie =O

f

aie

represents

Ee >

eo

such

an

element

that

implies

and,

by

the

Ay

nO

Schwarz

+

f m-a.e.

Indeed,

inequality

|h,, 2

by

Theorem

3.1,

h

h

he

>

O,s

=

of ff

n Ss f i

; This

by.

lemma. 5.2.

H if

.

indeed |)fll > [I7,l £1 ll 2 lItQEll - Hence [irl I> lel llas

s+o

Maxel,

(5 dy

m-a.e.

eite

oO

< IP" Ih Sa

ie

Let £rom:

(36)

such

that

rand"

m°

‘

llelk,

complete She

‘

space,

H,

it

follows

“asfunceron

ss

o,

‘

This s

implies: nu

iene

for

such

every

that

he

wa

represents

a

ny h

and

We

note

sequences

is

m°-a.e.

u_-

n

+e,

Hence

there iio

3 ny

the

that:

aes

es +O,

only

representative

of

h.

exist On

subsequences

chen

mss

93

et

Le

5.2.B.

An

To prove we

and

u

observe Je) But u

ly

F aly

ER

1

that h

On

element

5.2.A,

£

we

of

fix

gt Sage

the

other

OY

Ss

£ =

lim(h

hand

-h

an

product

inner

h

represents

ell

=

h

measurable

(we

do

in

Wf2

by

not

H a

and

(-A,f£,£)

represents

s,u

s

ee a!

h =

p > 0

Hence

le i u

that llell, < ©

ee

ty

Therefore

2

il

(m®) .

ceOgee lak:

@f.

such

(\£ - Ag yl P,P)

hy yl Te) SS

=SeD

We

TOMAS ners

conclude

f

.

the

f

that

and

the 4

h.

of

all

formula: Put

; is

f

which

=

(-A£,£)

4 meaningful

(h,h) W/Z

represent

Consider aie

for

every

jason

We

>|

Buel

There

woele

2 a

he

Tle

n

2

foe Tat)...

Te Et,

(S55)

teeeia

(5.6)

exists

S_, ray

B-

that

fopealil

Proof.

AE

i

elt = lim @,(f£) .

n

that

8.

functions).

(£,£)4)

lel), =

(1.6)

of

functions

m-equivalent

formula

© (£) = elt Soll,

set

by

represents

Note

function

H

distinguish

defined

2

a

> Tf

Ors

from 5.2.A and the definition

Denote

of

Da

-

hy se

) =h

s,uts

follows

5.3.

elements

sh

’

O,uts

5.2.B

then?

a function

San (\£ (

2

f-T

ih,

ITS l, < llelL, < 00,

(|Oni ie

(m®) .

cepresentse

>

an

element

f

m-a.e.

h

of

By

H

and

sequences

s

n

>

(3.6)

a9

5 lin|[e -£ 01h, = )napte |e = (0)

ei)

al =>

(Ga)

m->

and

(5.8)

el, = Wee,thy > Wolly = lelly By

Fatou's

lemma,

(5.5)

o, (£-£,)

and

= ilstim

(3.10),

shebe

Omit

te)

2

< eo

°

0,

94

Since

|e, (£) Si Since

2a

€

we

D ‘a!

have

Af ton

(5.6)

>

Af

follows

from

Remark

1.

(5.7)

It

; in)

n

1

2

i

and

9 and

every limit point of

By (5.9)

ll £,|

follows

(mm)

ay (5.9)

Hh

Be

‘

t+>0O.

as

Pe: Mtoe)

Las ie

lies between

t+0O

as

©, (£)

1/2

=o, (fe)

ale Me ale

®. (£)

Hence

1/3

form,

quadratic

semi-definite

a positive

is

o.

and

leatat ny formalin

and

(5.7)

=n.

(5.8). from

(5.9)

that

(5.10) 5.10

eylim 6, (£)). 0, (£) = noo

Remark

2.

eis

ae

We

have

-1

(Gi. a

the

proot

decreasing

of

and that

if

ell,

fo >

2

Sinee t

for

f

lI

tlh,

lA

c|lf||

in

H,

T,

and,

ele follows

m°|£| Hence

of

s)||£|\|

Sade

H

(3.10),

function

Qf)

F

JN PuyoFnlhse is

by

a

contraction

(5.10),

so

is

in

o,(£).

H,

of)

is

Hence

fe Ffrome

IONE

o.c7b.

GPS

a

clac

D-

for all then

(5.2) D

is

By (5.4).

and

everywhere

dense

in

(5.1)

fet.

(5.12)

subsequence

fn

converges

to

f m-a.e.

k Lemma EO

aytUuNncE ON

is

a bounded

=

5.2.

If

a sequence

then

tere | fis

It

is

well-known

sequence

in

an

Proof.

fo

(see

bounded

e.g.

in

[4],

ff

converges

Section

38)

m-a.e.

that

if

f

‘

a subsequence

our

case

the

f

ny

such

relation

E

arbitrary

that

> F

F.

k

in

=n, k H

Hilbert (fo.

ny

space

H,

4... 0+8.) ny

implies

that

then

there

converges

a

subsequence

in

exists H.

FL

j converges

to

F m-a.e.

Since

fF >

£ m-a.e.,

we

have

F,

k.4

>

f m-a.e.

and

In

95

£ =F

m-a.e.

Therefore

f ¢« H.

Corollary.

exists

a sequence

£

5.5. and

if

only

that

® .(£)

Cons

quently

ih

2

(Go)

2)

A function

¢€ D

n

is

Proof.

The

SiC

ror

(£-T

Gy Ss @) We

(5.2)).

x

aAb

the

5.2.

"only

zi

“‘elatel

Hence

ig

if"

So

part

dehy

+O

Sk

+

seigiel

f « meta)

follows

(G5),

as

from

(£,£)-(T

s+0O.

£f m-a.e.

ul

< -s

We

list

smallest

only

Gi

if

ane

n

belongs

Theorem

£,£)

=

Therefore

for

s+ . ao (Aq,

of

some

two

®, (£Ag)

This

follows

< LON A We

some

there

WEISln

to

H

if

5.1.

Suppose

so _(f)

SCS.

F° =s

Gf

f£f

in

F

S. => Op

jf £)au Aish

properties

numbers

say

EG ul

of

a

the

and

typ EEE,

G2.

functional

b

that

f

is

and

we

,-

We

denote

put

Indeed

Pe |

and

DOD

sulin is

fopsall

eH.

n

= 2c

).and

for

fAg

is

eh eo)

a'Ab')”

scOntracteLOn

1O




f m-a.e.

all

t

By Theorem

normal

Ge 2eOlm—anje..,

sew

:

|£(x) -£ (y) |
l[efin

is

Tf

an

=

and

almost

are

|£|

in

because

(5.13)

By (5.227. function

forall.

almost

fo >

r a Gy

excessive

£ m—aies,

g

that

implies

if

t.

(5204)

excessive

functions

and

if

< 6, (g)

an

almost

(Gs(f- T f,£)t

o, (£)

denote

by

excessive

function

f,

:

( 6.25 )

pokes

=

H

of

all

f£

belongs

the

set

(£,g-T,g)

< (9).

functions

representing

elements

ae

Theorem

is

(5.13),

say

o, (£)

estimate

= -(GAf, ,p) < |[-A£ || lbell,-

We

Lemma

f < g m-a.e.

of

the

= eal, > Welly sell, 5.7.

(cf.

of

prove

(f ,p)

We have

H.

version

almost

5.4.

excessive

and

Proof. and

f

is

almost Now

Taken

pes) Ov belongs

tO

is a normal

eine) Lemma

to

< 0, (£) 5.2.

excessive

by

5.2.A.

be

All

cle

to

of

to

524. Phe

H Put)

h ¢ ae

o, (£)

excessive

is and

The function

functions

Bya Theorem

It belongs

belongs

almost

loll, 1, let n n the martingale property, [Ly

By

convergence

exists

n

convergence

and by

is

theorem, either

the

lim

one

series

A

or

=

L

zero.

EEL

a =

=LL where L n jos pall n Oi" (L =O] and

exists

(almost

surely),

Let

measure

the

us

vo jItF 5a)

-

so

rapidity

It turns

out

j= that

each

random

us

agree

to

one

that,

term

of

variable,

to

if

say A

the

is

neglecting

precisely

the

the

series

possibly

sequence

finite.

a

set

is

set

nonnegative

infinite

of

The

of

on

a

random

theorem

ane

n20,

so

that

of

positive

variables

of

probability

where

set

Or

Kabanov,

zero,

the

converges

A

is

n2>0O,

Liptser

set

a nonnegative

probability.

converges

and

where

rapidly

Sirjaev

Lh 20

to

Let

one.

rapidly says

is

That

is,

B( be va, ||Rye ul

and

Here

and

Sirjaev

proof

being

we

result

present

in

motivated

its by

an

alternative

"martingale" the

desire

one

judge

to

proof

form, have

of

the a

the

search

simpler

Kabanov,

for

and

Liptser

a different

more

"natural"

derivation.

But statistical complete

how

does

hypotheses,

separability,

Singularity

of two

Le

Cam

which

measures.

has

are

(See

"naturalness"?

In

introduced

two

concepts,

the

absolute

analogues

of

Le Cam

(1960)

and

the

theory

of

testing

contiguity continuity

(1977).)

They

are

and and

concepts

101

which

relate

with

each

to

sequences

Consider

F

with

two

said

to

for

of

measures

which

are

not

necessarily

connected

other. a sequence

measures,

be

some

(n)

P

contiguous

sequence

and

to

of

of

(n )

Q

ph) y

events

A

measure -

The

if,

(Q.,A_), Ties

sequence

of

each

measures

endowed {Q

(n)

ees

whenever

¢« A,

n

spaces

then

n

ime oo! (A) = 0 noo“

also.

When

Q. =

0

woe

aulil

im,

ida

7

are

increasing

and

the

measures

n

P

(n)

and

An!

ce

(n)

Q

contiguity

PP

Ga

is

from

equivalent

to

measures

the

P

absolute

and

Q

anes

by

continuity

restriction

of

Q

with

to respect

n

On

are

SeESueAN

obtained

wW As

n

(o™)}

i

are

the

said eA

n

to

other

separate

such

n

hand,

sequences

entirely

if

of

there

measures

exist

a

{p!

n

)y

and

subsequence

n'

and

that

'

te

the

eee)

n'>0

to

=

while '

Gry‘

oe

n

ae

n-©

Again, and

the

if

Q

p(n)

on

a

and

single

of

P

and

Recently the

terms

of

Oosterhoff

absolute

are

it

van

to

)}

an

and

by

fo'™ 3

has

been

shown

that

continuity/singularity

Zwet

restricting

increasing

is

two

measures

P

sequence

of

o-fields,

equivalent

to

the

then

mutual

©.

contiguity/complete and

obtained

space

p(n

of

separation

entire

singularity

about

9”) measure

(1979)

separability. can

be

seen

a number

of

measures

Thus, as

for

of

classical

have

results

analogues

example,

a generalization

Theorem of

in 1 of

Kakutani's

Nye

theorem the

to

sequences

classical

measures.

dichdtony

It

therefore

Liptser

and

results

about

presented

Sirjaev the

here

van

First

and

we

)

gn

raat

with

F

to

be

of

OF

the

as

so

some

measures

=

n

ae

ee

and

define write

L

nj

|. =

dO

Oe.

eee

nj

i

h where

(ay

Sap nj

L Lees

a

an

Also,

a Oj(P

=

we

on

subscript random

measures.

such

Kabanov,

to

The

give

analogoug

proof

version

the

of

result

Suppose

the

of

for

theorem

of

Oosterhoff

each

and

n=1,2,...,

(Q0rA)-

when

it exists

increasing

and

sequence

oe

of

otherwise.

sub-g-fields

of

An

is

nj

well-defined

P

nj

( -a.sec.

:

| ayaawt

ih

i fe

pe

way Wetoh,

(HS

aes) Stns)

or

ea

*

O,

and

zero

i otherwise.

ao") ) ccopermareingale,

So

Note

that

that

b= nj

Bae

=

0;

= ae k=1,2,

as

soon

as

a

Onnon =

On

have

E

all

of

the

DERIVATIVES

notation.

on

which

nj

(n)

extended

See

._/dP., nj

L Sen

TSAR era pmol

E L oa é

for

=

nj

of

generalizes

of

include

(n) |F

Now

sequences

a proof

it

martingale

to

Li = ag'™ jap'™ be

if

sequences

RADON-NIKODYM

Sirjaev

two

to

judge

(1981)

that.

establish

are

two

Eagleson

measures

to

"natural"

generalize

j=1,2,...,n} =A

nn

Gaussian

whereas

reasonable

precisely

and

Set Let

between

seems

contiguity

shall

Lipster

Zwet.

(n)

measures

SEQUENCES

We

P

product

Theorem

does

2.

Kabanov,

of

oy tnglla je

and, on

the

variables

3:

(Where

expectation fo

< ot

no

confusion

operator.)

Da, 2 etn

if

pi”) (gl Se

(1)

oc

Sel

should Finally, On

GeO

arise, recall

we

shall

that

reral sequence

of

suppress

a sequence measures

the of p'n)

103

ese

Ne

etendsetosintinity;sunaftormmly

Theorem

1.

bounded

away

SS

Let

Bo

from

denote

zero,

a

n.

sequence

(n)

and

in

P,

of

(.)

A

denote

“measurable the

sets

conditional

with

P

(n)

(B_)

n

probability

n n

If) for ald”

p' Ais

ks O;

eae ee

lily

chensrona

js.

(n)

:

the

other

3

Px

4)

ok)

y

(Ly Se)

»

e= Ons

(3)

if

hand,

(n)

p

ullin Ga

Ts i

eight i

(eS

j=1

well

(2)

n

gaa (

as

SL,

ae

©;

lim n+

On

eve Gos

afaik

ig

n0o0

)

(4)

n

as

1)

max

Te

(Bey)

tight

being

F

(5)

n

j ce) j=l

noted

in

Oosterhoff

(8)

now

and

follows

van

Zwet

from

< lime poe

(1979),

-

fact

n A x ES (1 Seal) sata

because

one

vo He = ae

is

©

dealing

Ja

with

product

(g'™}

measures,

is contiguous

for

to

any

collection

of

measurable

sets

{A

n

.},

when

fo}

n ish neo

implies

OF 12 j=l

ONS)

SO)

that

n

iste

8 OL. GA.)

noo

3.

QO

also.

j=1

REFERENCES

Eagleson, G.K. An extended dichotomy theorem for sequences Gaussian measures. Ann. Prob. 9 (1981), 453-459.

Hall,

W.d.

and

Loynes,

R.M.

On

the

concept

of

contiguity.

of

pairs

Ann.

of

Prob.

5

(MIT) 75 ZHIFZEAs Kabanov,

Yu

M.,

Liptser,

R.S.

and

Vv

Sirjaev,

A.N.

On

the

absolute continuity and singularity of probability Mae. Sbornik LO4 (WAG) 1977)", 227-247. Kakutani, S. On equivalence (1948), 214-224.

Le

Cam,

L.

Carer Le

Local

of

asymptotically

weDUDIt meSictels

J..

Oosterhoff, distance. (Ed.

J.

Martingales J.

and

van

normal

families

measures.

of

Ann.

the

Math.

distributions.

49

Univ.

normality of estimates. Proceedings of the Neyman (Warsaw 1974). Panstw. Wydawn Nauk.

4 temps Zwet,

Contributions Jureckava).

product

of

(Russian).

Cemeom (L96O) Moy Ie.

Cam, L. On the asymptotic Symposium to Honour Jerzey Warsaw, (1977), 203-217.

Neveu,

infinite

question

measures

discret.

W.R. to

A

Statistics

Academia,

Masson

note

Prague

on -

et

Jaroslav

1979.

Cie,

contiguity

Paris. and

Hajek

(1972).

Hellinger Memorial

Volume

12

Oxford commemoration

ball

By J. M. HAMMERSLEY

1. Question for a demy. I hope this salute to my old friend David Kendall will remind him of his Oxford days, for it commemorates an excellent scholarship question that he once set in the 1950s when he was the mathematics tutor at Magdalen. The gist of the question ran as follows. A spherical ball of unit radius rests on an infinite horizontal table. You may imagine that it isa globe with a map ofthe world painted on its surface to distinguish its spatial orientations. The state of the ball is specified by specifying both its spatial orientation and its position on the table. You have to transfer the ball from a given initial state to an arbitrary final state via a sequence of moves. Each move consists of rolling the ball along some straight line on the table: the length and direction of any move are at your disposal, but the rolling must be pure in the sense that the axis of rotation must be horizontal and there must be no slipping between ball and table. How many moves, N, will be necessary and sufficient to reach any final state? The original version of the question, set for 18-year-old schoolboys, invited candidates to investigate how two moves, each oflength z, would change the ball’s orientation; and to deduce in the first place that N < 11, and in the second place that N < 7. Candidates scored bonus marks for any improvement on 7 moves. When he first set the question, Kendall knew that N < 5; but, interest being aroused amongst professional mathematicians at Oxford, he and others soon discovered that the answer must be either N = 3 or N = 4. But in the 1950s nobody could decide between these two possibilities. There was renewed interest in the 1970s, and not only amongst professional mathematicians: for example, the then President of Trinity (a distinguished biochemist) spent some time rolling a ball around his drawing room floor in search of empirical insight. In 1978, while delivering the opening address to the first Australasian Mathematical Convention, I posed the problem to mathematicians down under; but I have not subsequently received a solution from them. So this is an opportunity to publish the solution. However there are one or two surprises in store; so I shall not reveal until much nearer the end of this paper whether NV = 3 or 4, and in the meantime the reader may care to ponder which horse to back. Any mathematical paper ought to raise more unsolved problems than it resolves. So in §15, I shall mention some variants and generalizations of Kendall’s problem: most of these are very difficult and some may be quite beyond the reach

113

of contemporary mathematical techniques. Accordingly I call them problems for the twenty-first century. At the other end of the mathematical scale, §2 will deal with some nineteenth

century mathematics for the benefit of young mathematicians —undergraduates, clever sixth-formers (for whom Kendall set the problem originally) — and schoolteachers. Kendall’s problem raises some educational issues, belonging particularly to the twentieth century; for, both at school and university, the pursuit of abstraction and generalization have come to vie for time and attention against the development of manipulative skills.To handle rotations, there are two main tools: orthogonal matrices and quaternions. The former generalize to n dimensions, but the latter win hands down in the particular case n = 3 when it comes to manipulations of the sort needed in Kendall’s problem. Although undergraduate lectures devote much time to vector analysis in mechanics, scarcely any newly—fledged honours graduate nowadays knows what is meant by the quaternion product of two vectors or how to represent rotations by quaternions. Indeed, if undergraduates have met quaternions at all, they may only have met them as a passing illustration of anon—commutative division ring; and that is a sad reflection on modern education. So I do not hesitate to describe quaternions in §2, in a thoroughly oid—fashioned style, as a valuable manipulative tool. The first paragraph of §2 summarizes the conventions, notations, and results that will be used in this paper; and readers, who are familiar with quaternions, will not need to read the remainder of §2. Other readers, who wish for an introduction to quaternions, should not be deterred by that first paragraph; for they will find explanations, definitions, and proofs in the remainder of §2. I shall assume that all readers are familiar with the elementary facts of vector analysis; and these prerequisites appear in the second paragraph of §2 for the benefit of clever sixth-formers and schoolteachers. Anyone, who understands the second paragraph of §2 (supplemented, if need be, by a textbook on vectors — for example the first nine pages of D.E. Rutherford’s Vector Methods (Oliver and Boyd, 1939)), ought to be able to understand most of this paper, except perhaps §§13 and 14 which are more advanced.

2. Nineteenth century prolegomenon on quaternions. We shall work in three—dimensional Euclidean space referred to a right-handed Cartesian co-ordinate system, with mutually perpendicular axes Ox, Oy, Oz of which Ox and Oy are horizontal and Oz is vertically upwards. Bold-face letters denote vectors in this space, and the corresponding italic letters denote the length ofthese vectors: thus v is the length ofthe vector v. We write i, j, k for unit vectors along the axes Ox, Oy,Oz respectively. The letter u (with or without suffices) is reserved to denote a unit vector; the letter h (with or without suffices) is reserved to

denote a horizontal unit vector; the letter g (with or without suffices) is reserved to denote a quaternion. The quaternion product of two vectors v and wis written vw (without a dot or cross, to distinguish it from the scalar product v.w and the vector

114 product v x w). The angle 0 between two vectors vy and w is the angle between their directions, and is always understood to satisfy 0 < 0 < 2, where @ =

1 if and

only if vand w are in opposite directions. If u is perpendicular to both v and w, the angle from v to w along wis the angle from the direction of v to the direction of w in the plane of vand w measured clockwise when looking along the forward direction of u, such an angle being interpreted modulo 27; and, conversely, use of the phrase “from vy to w along u” will automatically imply (without further explicit mention) that u is perpendicular to both y and w. A rotation through @ about u means a rotation about an axis u with the angle of rotation @ measured clockwise when looking along the forward direction of u. This rotation is represented by the quaternion pair +q = +e" = +(costp + usintg) = +uju, , (2.1) where the angle from u, to u, along uis 4¢. Angles of rotation may be interpreted modulo 2x when the context permits. Conversely, if +q, and +q, represent rotations whose axes are perpendicular to u, then

+1 4 +e?" = +99,

(2.2)

implies that +g, = +u, (p = 1,2), where +o is the angle fromu, tou, along u. A sequence of rotations, represented by the pairs +q,,+4).,...,+q, and performed in that order, yield a resultant rotation represented by

EQ =

£4n+++4291

-

(2.3)

We shall assume the following prerequisite knowledge about vectors. We shall always use the noun scalar to mean a real number. The vector v = (x,y,z) is the directed line from the origin to the point having real Cartesian co—ordinates x, y, z;

and thelengthofvis

v = ,/(x* + y*+2z’) > 0.

The vector visa unit vector if

v= 1. Let w= (é,n,0) be another typical vector. Vector addition is commutative: v+w = w+v = (x+é,y+y,z+). A vector can be multiplied commutatively by a scalar:av = va = (ax,ay,az). The scalar product of yand w isv.w = w.v = x€ + yn + 2¢ = vweos8@, where @ is the angle between v and w. The vector product of v and wis vxw = (yC — zy,z€ — x€, xm — yé), which ‘is a vector of length ywsin@ perpendicular to v and w and has the direction of uwhen @ is the angle from v to w along u. Thus vx w = —wxv. All the ordinary laws of algebra apply to the addition and multiplication of vectors, provided that due care is taken over the order of the factors in any vector product and over the bracketing of terms in the triple products Wex Wick = Ve(wooth, (2.4)

vx(wxt) = (v.t)w — (vew)t ,

(2.5)

(vx w)xt

(2.6)

=

(y.thw

—

(w.t)v.

With the aid of the foregoing preliminaries, we can now define quaternions and derive their properties. For conciseness, we shall not repeat the conventions and

115 definitions above, to which the reader may refer as and when he needs them. A quaternion is a combination of a scalar and vector; and, for reasons of manipulative convenience, we write this combination in terms of commutative addition: q=a+v=veta. (2.7) Here ais called the scalar component of q, and vis the vector component of g. Two quaternions a + v and b + w are equal if and only if a = b and v = w. Scalars and vectors can be considered as particular sorts of quaternion by taking v = 0or a = 0 respectively, giving g = a or q = v. The addition of two quaternions is defined by the addition of their components: Sea)

Seoala Lah CGS ed here AEE ehDc

(2.8)

Thus quaternion addition is commutative. The quaternion product of two vectors v and w is the quaternion with — v.w for its scalar component and vy x w for its vector component; and it is denoted by vw

=

—V.w+vVxXwW.

(2.9)

More generally the product of two quaternions is defined by

(a+v)(6+w)

=

ab+aw+ bv + ww (ab — v.w) + (aw + bv + v x w).

(2.10)

It follows, by a straightforward calculation from (2.4), (2.5), (2.6), (2.9) and (2.10), that the quaternion product of vectors is associative, (vw)t = v(wt); and hence from (2.10) that quaternion multiplication is associative, (¢,q2)¢3 = 41(q293). (Note that vector multiplication, as instanced by (2.5) and (2.6), is not associative: to this extent, quaternion algebra is easier to handle than vector algebra.) Quaternion multiplication is non-commutative, because vxw = —wxvV. However, according to (2.10), any scalar gq= acommutes with every quaternion under multiplication. The conjugate of the quaternion q = a + v is defined and denoted by g = a — v. Hence wy =

—Vv.W

—

VXW

=

WwW;

(2.11)

and (2.10) and (2.11) now show that

Moreover, if g =

a+,

9192 = 92% -

(2.12)

qgq@=q=a+wv230,

(2.13)

then

with equality if and only if q = 0. Since gq isa scalar, any non-zero quaternion q

has a reciprocal qg~! = q/(qq) such that qq°' = q-‘q = 1. Since scalars commute with quaternions, (2.12) gives

(4192)(4192) = 41(4242)41 = (4191) (4292) -

(2.14)

116 The quaternion q is a unit quaternion if qg = 1,in whichcase q_' = g. According to (2.14), the product of two unit quaternions is a unit quaternion. Hence the unit quaternions form a non-commutative group under multiplication. All the ordinary laws of algebra, including division by any non-zero quaternion, hold for the manipulation ofquaternions, provided only that the order ofthe factors in any multiplication or division is respected: in technical language we express this by saying that the quaternions form a non—commutative division ring Q. In this system, each unit vector is a square root of —1, because u? = —u? = —1, by (2.9); and

the converse holds as well, since g? = (a+ v)? = a? — v? + 2ay = —1 implies v2 = aq? +1 > 1and2av = 0, whencevy ¥ Oanda = Oandv? = landq =v must be a unit vector. As a consequence of(2.13) and the fact that scalars are real, we may write any unit quaternion in the form

q = costo + usinig = e2% ,

(2:15)

for some angle g and some unit vector u, upon expanding the exponential formally as a power series and usingu? = —1. This exponential is to be thought of firstly as a convenient shorthand; and, since quaternions do not commute, we cannot write eve¥ = eY+¥ unless v and w are parallel. However, the notation (2.15) is much more than a shorthand: it is a powerful manipulative asset in more advanced situations, such as the solution of quaternion differential equations. The reason for inserting the fraction $ in (2.15) will become apparent presently, when we consider rotations. We now identify the quaternion q in (2.15) with rotation through @ about u, as defined in the first paragraph of §2. But, since this is the same as a rotation through g + 27 aboutu, the quaternion —g = cos3(y + 27) + usin3(p + 27) is also identified with this same rotation. Accordingly we say that the quaternion pair +q = +e? represents a rotation through g about u, as stated in (2.1). To prove the remainder of (2.1), consider any two unit vectors u, and u,, each perpendicular to u, such that (in the language defined in the first paragraph of§2) the angle from u, to u, along u is 4¢. By (2.9) u,u,

=>

—u,.Uu,

=

—(u,. u,+u, xu,) =

+ U,

x

Uy

— (costo + usin49),

(2.16)

and accordingly +u,u, is the representative pair of quaternions for the rotation through @ about u. On the other hand, given the foregoing vectorsu, u,,andu,, considera sphere S freely pivoted about the origin O and subjected first to a rotation through z about u, and then toa rotation through z about u,. These rotations have representative pairs +u, and +u,,asweseeon putting @ = 7m in(2.15). Let Cbe the great circle on S passing through the ends of u, and u, ( or any such circle if u, = +u,). Each ofthe rotations is a rotation through z about a diameter of C; and therefore each point of C is transformed into some other point of C after both rotations are complete. So the resultant rotation is a rotation through some angle

117

@ about u. To see what 6 is, consider the point of C initially coinciding with the end of u,. It remains fixed under the first rotation, and under the second rotation it moves to its reflected image in u,. Hence 6 = 2($¢~) = g. Thus the resultant rotation is represented by the pair +q = +u,u,. In other words, we have proved the particular case of (2.3) in which n = 2 and q, = u, and gq, = up. Next’consider the somewhat more general case of(2.3) in whichn = 2 and also 1 1 : ° q, = e274) and g, = e2?242. Let u; be a unit vector perpendicular to both u, and u,. According to (2.1), we can find vectors u, and u., such that5@, is the angle from u, to u, along u, and 4@, is the angle from u, to u, along u,; whereupon +q, = +u3;u, and +q, = +u.u,. Let +g = +e2%" be the resultant rotation from performing the rotations represented by +g, and +q, in that order. From the special case proved in the previous paragraph, this is the resultant rotation produced by rotations through z about u4,u;,u3,u, in that order. Since two successive rotations through z about u, together preserve the orientation of S, the net resultant rotation represented by +q is the resultant of rotations through z about u, and u, in that order, which by a further appeal to the special case proved in the previous paragraph, is represented by +g = +u.U4. However u3 = —I1 commutes with any quaternion. Hence +q = + u,u,u3U, = +4 q,; and this proves (2.3) form = 2 and arbitrary rotations represented by et qi

+42.

We can now prove the general case of(2.3) from the particular casen = 2. Since quaternion multiplication is associative, we put +g = +49,(4,-1---9291) and, by induction on nv, use the special case to combine the rotation represented by +4,—1---2q, With its successor represented by +q,. This completes the proof of

(2,3): Finally, to prove (2.2), we may write it in the form +1

4

+(c+su)

in which c = cos$g,

s = sing,

cannothaves = 0,forotherwisec each side of (2.17) we have

su

=

=

+(c, + 52u,)\(c, + 5,U,),

c, = cost, =

(2.17)

Ss, = singg,

(p=1,2). We

+1.So, equating the vector

components on

+(c25,U, + C,S2U2 + 54S2U2 X Uy) .

(2.18)

By hypothesis u, and u, are perpendicular to u. Resolution of (2.18) along u and perpendicular to u gives su

=

5,S,U5

x

u,

3

C2S,U,

+

€,S2U,

=

0

.

(2.19)

Thus s, # Oands, # 0, for otherwise s = 0. Hences,u, and sju, are linearly independent vectors; and the second equation in (2.19) now implies c, = c, = 0. Hences,

=.

+1, ands,

=

+1.Sot+¢,=

14,

(p=1,2), as required.

Quaternions were used quite extensively in nineteenth century research: for example, Maxwell originally wrote his equations of electromagnetic radiation in terms of them.

118

3. Necessary and sufficient conditions for an N-move solution. We may suppose that, in its initial state, the centre ofthe ball is at the origin O of our fixed co-ordinate system; and, in its final state, the centre of the ball lies at the end of a prescribed vector v. Since v is horizontal, it is perpendicular to the unit vector k in the upward direction; and hence — vk = k xv is also a horizontal : vector, which we can write as —vk = Oh. (3.1) For the pth move (p = 1,2,...,N) we suppose that the ball is rolled through an angle y, about an axis that has the directionh,. Here , may have any real value, positive, negative, or zero; and |@,| will be the length of the pth move. The vector gph, completely represents this move; and, for brevity, we shall simply call it “ the move ¢,h, ”, although actually it will shift the centre of the ball by a displacement vector y,h, x k = y,h,k. To bring the centre of the ball to its desired final position, we must have N

»y gy,h,k

=

v ;

(3.2)

p=1

and multiplying this equation on the right by — k, we get N

Seo

hes on.

(33)

p=1

To attain the desired final orientation, the ball must undergo a prescribed rotation,

represented by + e294 say. Hence e2Pnhy ae exP2h, ere ih,

=

+ ezeu

:

(3.4)

To achieve an N—move solution, it is necessary and sufficient to be able to solve

(3.3) and (3.4) for the N horizontal vectors ~,h,, given any prescribed vector gu (not necessarily horizontal) and any prescribed horizontal vector 0h. This includes the possibility that fewer than N moves might be sufficient in some particular case: for example, if the initial and final states coincided, no move would be required and we could satisfy (3.3) and (3.4) by taking y,h, = 0 for all p. We say that the pth move is integral (or even or odd) if ~, is an integral (or even or odd) multiple of z; and we say that we have the integral (or even or odd) version of Kendall’s problem if we require at least one move to be integral (or even or odd). We write Nintegrals Neven» Noda for the necessary and sufficient number of moves in these respective cases. Notice that a null move (i.e. one with g¢, = 0) counts as an even move. Generally, if Nprescribed 18 the necessary and sufficient number of moves when at least one move (independently of other moves) must be of some arbitrarily prescribed kind, we have i NS

Nprescribed

=
1, such a solution is not a convex surface. I am not aware of literature on the convex solutions of partial differential equations satisfied almost everywhere; and this topic merits further investigation.

14. Quaternion calculus of variations. In this section I shall examine how the calculus of variations may be applied to problems (a) and (b) of §13. Standard techniques seem to give a nonsensical answer, but an adhoc parametric technique is more fruitful. We shall roll the ball along the curved path I on the table, starting from the origin and writing ¢ for the arc length along [ measured from t = 0 at the origin. We write g = g(t) for the quaternion such that +q represents the resultant rotation of the ball at t. We take q(0) = 1 and suppose that g(t) is a twice—differentiable function of t. We use dots to denote differentiation with respect to t. Leth = h(t) be the unit vector along the instantaneous axis of rotation at f; so, in rolling from ¢ to t + dz, the ball rotates through dr about h. Thus

q(t+dt) = ethdt g(t) = {1 +4hdt + o(dd}q(0);

(14.1)

whence

q =

hq.

(14.2)

dq = fot fhi+Ait+ sk,

(14.3)

Thus

th =

where thef, denote the quaternion coordinates of Sh. If the coordinates of g are

q = Pot pilt paj + p3k ,

(14.4)

then (14.3) can be written explicitly in the form fo fi fo fs

= = = =

— BoPo —BoP1 —BoP2 —BoP3s

+ + + —

B1P1 61P0 6103 6102

+ — + +

62P2 P2P3 620 62P1

+ + — +

P3P3 P3P2 6301 P3Po-

(14.5)

Since h is a horizontal vector, we have the constraints

fo =f, = 0

(14.6)

identically for all ¢. Also in problem (a), though not in problem (b), Jo hdz gives

the prescribed final position of the ball on the table; so [3

f,d¢ and [7 /, dr are

also prescribed. We need to minimize T = {6 1 dt, subject to these constraints and subject to a prescribed final orientation q(T). First variations will not alter a stationary value of T; so, for first variations, we can treat Tas a constant. Thus we

look for stationary values of (6 fide , where

f =

14 dofo + Ath + Aafy + Asfs.

(14.7) Here the 4, are Lagrangian multipliers; 49 and A, are functions of t, while

139 24, and taking The M. R.

A, are constants. We can cover problem (hb) as well as problem (a) by A, = A, = 0 in problem (6). foregoing procedure is standard practice (covered in Theorem 5.1 of Hestenes (1966) Calculus of Variations and Optimal Control Theory, p.

263); but unfortunately it leads to a contradiction. I do not understand what has

gone wrong, despite consulting one or two experts in the calculus of variations. Perhaps the reader can locate the cause of this trouble. The Euler equations

of —

op,

=

dif

of ;

at fe sf = 10M eras)

become in matrix form

‘

Po

Pi

“Paw

ae.

Ps

[—4o|

Pie

Po

Ps

2

Pz,

Sia

Pou

Ps"

a14.8

aPraiPi,

=

0

P2

2A,

0

iw Pa

213

0

FePo —Po

2h3

0

(14.9)

A3 Taking norms in (14.3), we have

(14.10)

qq = |

4 = 4449 = 44. and hence, on premultiplying (14.9) by Pam

Pow

PIES

Gil

“a

Psat

P23

Po

FP1

PonLe3 mile 02

bo —Ps —Bo bi

and using (14.6) and (14.10) we get

a=

0

bh

O°

WO

Sh

-0

sl

0

0

OR

i,

0

244

0

2h>

0

ifnoe ge

A

Eo

Ot

ONE,

Mahi= Wo = faire oas

(14.11)

0

Ae The first two equations in (14.11) yield Afi

=

“a2

=

2:

But, according to (14.3) and (14.6), h is a unit vector; so fp (14.12) now implies 24; = 0 and thence 4, = 0. Thereupon

(14.12)

and fee the last two

136 equations in (14.11) give

Ake = T4AL,

,Aole. = te

So 442 = 42 (7 +6) = 42? + 23), which:

(14.13)

is constant.

If 4g =70,

then

(14.13) provides 2, = A, = 0, and the Euler equations become vacuous.

So

AG must be taken as a non-zero constant, and then

h = (i+ A,p/Ao

(14.14)

is a constant unit vector, and I can only be a straight line. To circumvent this stumbling block we pursue a parametric argument as foliows. We write 8 for the set of quaternions having the form q = a+ bk, where aand bare scalars. Since k a fixed square root of —1, wecanregard ® asan embedding of the complex field in the quaternion division ring; and we can treat the members of 8 as ordinary complex numbers. Accordingly we rewrite (14.4) as

G2=*

Gatigol

dy =.

Suppose h subtends an angle

06 + Pak EK, YW =

“Que =o

Pp; + pak “ern

4)

w(t) with the x-axis:

h = icosy+jsiny

= e¥ki = ie—¥k.

(14.16)

The curvature of I” at the point ¢ is

k = wW.

(14.17)

Gi t+ Gai = e¥Ki(g, + gai) = 4e¥K(—G, + Gri),

(14.18)

Equation (14.2) becomes

in which we may equate the pre-coefficients of i:

di: =

—4e¥KG,,

dy = 4e¥KQ,.

(14.19)

From (14.17) and (14.19) we deduce that g, and q, are both solutions of the differential equation

G, —kkq, + 4q, =

0

(p= 1,2)

(14.20)

with respective boundary conditions

9:0) = 1, 4)

= 0; (0) = 0, 4,(0) = Fe¥Ok

(14.21)

obtained from inserting g(0) = 1 into (14.19). Moreover, because q is a unit quaternion, there exist real functions a = a(t), b = b(t), c = c(t) such that

q, = eKcosda,

=gq, = e2Ksinda.

(14.22)

Substituting q, from (14.22) into (14.20), we find

(1 + 2bx — a? — b? + 2bk)cos4a + Adwk — 4 — dbk)sinta = 0.

(14.23)

The real and imaginary parts of this complex number must both vanish. This gives

137 two equations, between which we eliminate x to get

(1 — a*\(dsinacos 4a) — 2dasin?4a = 2bbcos?4a — b2dsintacosta;

(14.24)

and this integrates to

(1 — d?)sin?4a = b*cos*4a + constant.

(14.25)

From (14.21) and (14.22) we have

a(0) = b(0) = (0) = 0;

(14.26)

so the constant in (14.25) is zero, and hence

b = «(1 — d’)r tanta where ¢, =

(14.27)

+1. Similarly, substituting q, from (14.22) into (14.20), we get

é€ = 2,(1 — d?)2cot4a with ¢, = +1. We can now corresponding to p = 1,2:

Kane

calculate

(14.28)

« from (14.20) in two different ways

td) (1 —a*)icotale= \— 2, {a(l’9d2)-? =a)

cota. (14.29)

Hence ¢, = —é,. Next integrate (14.27) and (14.28), using the boundary conditions (14.21) and (14.26), and insert the result into (14.19) to obtain

aO)eYE = {4+ k(1 — d?)2}exp{ky(0) + ke, |(1 — d?)cotadt}. (14.30) 0

Then differentiate (14.30) logarithmically in a sufficiently small neighbourhood of t = 0 to cater for the principal values, and compare the result with (14.17) and (14.29). This gives ¢, = +1. We can also assume without essential loss of generality that ad(0) = +1; for otherwise we can simply replace a(t) and W(0) by a(—t) and y(0) + 2 respectively. Hence we obtain the required parametrization t

gi) =1C0S83a exp(— 4k |(1 — d?)2tan4adz}, 0

(14.31)

[

g, =

singa exp{ky(0) + aK |(1 — d?)2cot4adr},

(14.32)

10)

—hi

=

eVK =

{4+k(1 — d?)2} exp {ky(0) + Ka a ay cotadt}. 0

(14.33) where (0) is an arbitrary parameter and a(t) is an arbitrary parametric function for t >

0, satisfying the boundary conditions

a0) = 0,

a0) =

1,

Sls

dos

ul.

(14.34)

138 When the ball rotates through d¢ about h, its centre suffers a translation hkd?. So the final position on the table will be T

Ts

(14.35)

(7) = | hkdr = | eVKik di. 0

0

When the final orientation g(7) is prescribed, then a(7), 5(7), c(7) are prescribed via (14.15) and (14.22). So we require

(14.36)

(T) = - | — @)2tantadt, © 0

(14.37)

c(T) = 2y(0) + ix — d?)2cot4adt, 0

v(7)j = |,efar

(14.38)

0

We can secure (14.37) by choice of the arbitrary (0), whereupon (14.38) becomes

v(T)je—2(Dk = |,efter,

(14.39)

0

where LE

t

'd— s| (1 — d?)tcot4adt + |(1 —d?)2cotadr.

g = cos

0

(14.40)

0

[§ 1 dt subject to (14.36) and (14.39). For problem (a), we have to minimize T = So, introducing constant Lagrangian multipliers 4 and yo, where y is real and o is a complex number belonging to R, we have to minimize the real part of fe

i=

| 1— pl — d*)z tan4a + poeskdi

(14.41)

0

for variations in a(t) subject to (14.34) and prescribed a(7), where T is constant. Hence the required Euler equation is

1d(

tanta

adil —@p

oo

Oy =

Oa.

—

dd —-

aha

(deed eo) —

2

amen =

eo dé = RiiS — — —— ay See]oesk:

(14.42)

in which ® denotes the real part of the subsequent expression. This is a non-linear integro—differential equation, which looks forbidding; and I

do not know how to solve it analytically. Of course, it can be written out explicitly by substituting (14.40) into the right-hand side of (14.42). It could then be solved numerically, though this would be quite a formidable task even on a large computer because o and T are adjustable parameters to be chosen so that the solution satisfies the constraints (14.36) and (14.39).

139 On the other hand (14.42) easily yields the solution for problem (b), for which there is no constraint (14.39); so we may take o = 0. Thus (14.42) guarantees that (1 — d?)-2tan4a must be a constant, say 1/y. If 1/y = 0,thena = 0 forall t. This corresponds to the trivial case in which the ball remains at the origin because its initial and final orientations are the same. Setting aside this trivial case, we have

(1—d?)i = ytanta,

(14.43)

a(l — d?)-2 = —4ysec?ta,

(14.44)

and hence

Substituting (14.44) into (14.29) and recalling that ¢, =

+1, we obtain

K = 4ysec*4a + ytansacota = y.

(14.45)

Thus the curvature of [ isconstant,and I must be the arc of the some circle or a

segment of a straight line. This confirms the result already obtained for problem (6) in §13. When x is constant, (14.20) and the initial conditions (14.21) yield an explicit solution for q,(t), whose real and imaginary parts provide 2amcos4acos4b 2wmcos4a sin4b

(@ — k)cosdh(@ + k)t + (@ + k)cos#@ — k)t

nkw — x)t, (14.46) (w — k)sind(@ + K)t + (w + K)si

where w =" ,/(1 + x?) > 1. Squaring and adding these two equations, we get

wsinta

=

siniot

(14.47)

after taking a square root, whose ambiguity of sign is resolved by conformity with a(0) = +1in(14.34), Remembering that T > 0 by hypothesis, we can now define a and 8 by means of

@ 1 = sine,

k/o = cosa,

0 0. (14.48)

psing:

(14.49)

Hence

Ke=aCOUCm

lee

To complete the solution of problem (5), we must express the initial direction, the curvature, and the length of I in terms of the prescribed final orientation q(T) = costacos$b + isinjacos4c + jsintasintc + kcostasin}b. Here and hereafter, for brevity, we have written a = a(T), b = b(T), c = c(T) for their prescribed terminal values. If we change (a,b,c) to (a + 22, b + 22, c + 27) or to (—a,b,c + 2n) or (a + 27,b,c), we either leave g(T) unchanged or else change its

sign. Since +g(T) both represent the same final orientation, there is no loss of generality in imposing the terminal conditions

On< dal< duu 0 < dosenals den

recurrent

it

(1)

to

m when

Q

is

transient

linearly

condition

be

conservative,

space

S,

transition

Markov

and

regular

probabilities

process

and

let

determined

of

the

by)

9Q

(1)

©

is

Q

5 eS)

countable

(meyds © S), Uundque sede for P(t) if

Q

sufficient

process.

generate

sufficient

seers =

When

it may

to

and

relations

) mt, peAt)

or

necessary

=O

J

invariant

a

€ S)

ma.

is

a

a Markov

ee

over

) ma,

an

for

assumed

q-matrix

(py

Markov

provides

introduction

irreducible

P(t)

.

paper

for

is

is

to

constant

multiples.

Call

em

0,

-o,

then

by

given

be

and

J) s, 6, (x)£(x)

iby)

GO

5

(4.2)

p.74.

convenience

f(x)

let)

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1

for

XSK

(it

must

a non-

equal

*

zero

constant

a piecewise

stant

by

linear

function

Ge,4

(4.1)

boundary

jump

+

f= ri

function

equal

function

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condition).

continuous

with

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the

f

to

can

a (x, xe ace ) if

-2 be

and

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derivative

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aL

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that

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implies

oe

on = i

f.

and

in

Al

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ers

—)

A

-

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2

formula

in

terms

of

f foot =

Comparing

2

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s,f,

if

Brownian

function

(4.5)

with if

a

+

=

ee

the

{£,},

values

{£5}

pitas (2.4)

we

£

+

derivatives

(ib ap ich Yc! alt sli 1

P,(s)

;

the

the

and

polynomials

defined

the

(4.3)

+

Stay

Eliminating

Thus,

ifs

=

general

Re

the

(4.3)

fi

eas

we

can

get

birth-death

a recurrence

alone,

Pie Ad Ya see

. (24)

that

Hi

the

process

aes

{£3

has

nea

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birth

the

and

(4.5)

same

death

as

rates

by

the

starting

motion

has

corresponding

point

the

same

birth-death

a

equals

finite

one

of

dimensional

process.

In

the

points

sojourn

particular,

if

Kes time

then

the

distribution

asx,

the

mgf

as

170

op 's) using the

can

be written

Theorem

3.1.

tri-diagonal

in the

Since

the

matrix

dsp 8)

Q-matrix

Sih

2

form

is

sal -1 det (sr ~) /det (42

=

defined

given

by

(4.6)

already

+ Ss.) symmetric,

by

(4.7)

Remarks

(1)

occurs

if

the

boundary so

(2)

at

a2r.)-

ro But

the

next

bt+x)

for

motion

has

where

eo

we

shall

section a>b

O

4.a

32 SHE

1.

1.

The

and

hence

Cy

is

a

+ a 2a

By

left

it

SeONde 1

PIR TO ~ (f£, +b fy)

ap ee 7.

4a fF

> O

homogeneity,

there

is

hand

a non-decreasing

side

suffices

equivalent

(fore 6'4 17% See

Shaey

sae ( (Ga 3 a7 Zhe 1'2

aan

is

to

to

the

ee oe

prove

no

the

of

inequality

assertion

2

loss

that

ee

. pha (Ne

whenever

f£ 20,

how

to

his

proof.

prove

a@ >

(1.9),

Thus

2

and

C

numerical as

n

oS

No

I am

and

C,

proof

experiment

will

is

is

(ES 9) aie

indebted not

suggests

of

to

spoil

true,.and

known

ae)

so

any

that

John

the

reader's

likewise

Ch they

Hammersley

for are

showing

enjoyment

(but

more

n 25. true,

for

but

by

revealing

easily)

~Avcertaim become

me

are

Cy

amount

more

of

delicate

increases. For

with

1.

ati

7

> O,

a particular (1.9)

holds

by

renewal

sequence

continuity

for

and

a particular

oa -

1

value

sufficiently

of

small.

n The

183

same

will

be

Expressing

true

this

even

(ee

this

means

if

fn —ROmEsomlongeas!

derivative

that

in

terms

of

the

weak

that

it

version

the

is

of

strict (that

trivially

true).

considered,

Tati

=

ale

derivatives

Glo}

sufficient

(1.11)

is

inequality

sequences

sO

n

3g / 3x, ,

that

pou

The

dow /due

partial

is,

(1.11)

sequences

Hence

which

necessary

has

is

with

Cn

fact

true

une

sequence

advantage

of

holds

any

5

for

a

a companion

the

in

EAN)

(lester)

to

of

but

except

for

being

hold,

for

which

the

conjectures

free

of

the

it

seems

likely

geometric : conjecture Se

might

parameter

is be

a.

*

C_:

The

inequality

sequence

with

What that

Oe

Se

is

sel 2s

tsue

for

ali

do

obvious

together

with

sequence

negating

(1.7)

the

fails.

us

=

are

anduction res

this

mn,

by

Then,

of

and

~n) that

by

Crs

let

that

be

To

generalised

renewal

cs

oa

conjecture,

on

this the

etl

(u)

be

*

that

4

os

infimum

cmmc

implication,

assumption

nis) (esa bere

assume

the

original

Nconjunctvongo

establish

to) prove

Let

8

the

bys tne

vacuous).

on

“and

is

“mplied

contradiction,

negation

0

non-geometric

Sei

perhaps

and

(by

fom

To

which

rae

not

ie,

SsuLrererent

Tse

ON

holdsmtocraliaane

imgone felieil Vests

(1.11)

f. =

(Ge

1

for

Cee

*

Define

=

iu)

generated

to

*

(a)

= ©

by

(m > n)

(1.1)

to show

with

that

and these

Vet

A fa

do /da >O

(u_))

be

Then

at

the

*

generalised

Beye

a=

8.

g

Ore

Hence

renewal)

Se St,

lintel

sequence

*

oe

applies

184

-

a

Swith

18)

(n>)

for

In other

words,

the

conjecture

of

[7]

will

be

*

we

can

prove

established

*

ee

fOr

alles

enh Ses

tclce

KMOWn

etches

definition

the

contradicting

small,

sufficiently

8

if

*

C3

and

CA

are

true,

in

[5],

and

concerns

of

continuous

*

but

Ce

remains

2.

The

undecided.

Markov

This a Markov

Pi

: [O,;°)

=

group

problem

semigroup

on

was

the

[O,1]/

_

posed

by

countable

indexed

peaie (0) eee Ble j

problem

by

Kendall

set

2,3

S,

€ S,

himself an

array

which

functions

satisfies

Do eather kL,

5

cs nE ica

(7 ail)

fovt)) t) B= +

p,,(s

Lee

Py,

£) (¢)

keS The

definitive

account

The

of

of

semigroup

contractions

on

the

called

of

course

because

2 = & (Ss)

of

it

the

book

determines

sequences

x =

by

a

Chung

[1].

collection

(x, pak ene) | yysuiele!

:

formula

and

these

One

a

in

form the

P

Kendall

so

is

9 ia

1éS

the

is

systems

space

IES eae by

such

noted

a

strongly

sense

=P

that,

continuous

semigroup

of

(bounded

linear)

operators

that

Po(s,t,

if

the

2 0), lin tyo

semigroup

I|P x -x|[=o.

is

uniformly

continuous

(2a

in

the

sense

that

Lino tO

a

Ore

:

(2.4)

185

Ieebeingethe

si dentity,

P,

for

a bounded

and

that

=

exp

then

operators

on

only

(2.4)

2.

It has

Davidson who

[3]

proves

condition

that

(2.5)

the

be

makes

semigroup

conjectured

Markov

been

group

the

a most

as

written

that,

sense

to

as

even

for

a strongly

conversely,

conjecture a

bounded

subject

Cuthbert

of

[2],

is

negative

continuous

P,

can

equivalent

kinear

research

but

ingenious

be

t,

group

so

of

extended

the

by Kendall

best

argument

[5]

to

on

&,

operator

[5],

assertion

then

(2.4)

Speakman

[11],

result

is

due

(2.4)

is

implied

that

the

to

Mountford by

[1o],

the

that

pice The

2,

invertible

and

by

can

holds.

sie Po is

holds.

P.

(25>)

extends

He

This ANE,

then

(tM)

operator

(2.5)

when

on,

inequality

oad,

(2.6)

(2.6)

implies,

but

is

not

implied

by,

the

invertibility

of

pe

At discrete nature

this

problem of

is

really

of

the

the

parameter

finite

discrete

Markov

to

the

led

disjoint

semigroups. has

it

may

since t.

reasonably (2.4)

In

it

can

matrices,

finite

Thus

on

that on

be

that

the

argued

simplest

this

the

of

is

not

the

conjecture

all

the

tools

this

conclusion

Take

as

S

by

trying

a countable

to

construct

a

counter-

union

(2a)

sets,

for

a

continuous

enthusiast. to

each

and

n

off-diagonal

for

P.

let

the

OF

direct

be

elements

a typical

element

of

and

&%

product

a Q-matrix

exp (tQ.) = (py, (t) > Ly eescy P.

objected

crucially

US

non-negative

Define

be

depends

fact,

stochastic

conjecture.

Saaa

of

all,

about

I was

example

stage

at

zero

row

3

written

on sums),

of

finite

Sh

(so and

Markov

that

On

let

(2.8) as

186

See

(Ge

pelle

ely, ec erors) n

by (ex

=

); ieS

J

and

(2.9),

and

n

the

(GBP as

result

is

sup ||exp (-9) || hoon n

is

where

the

matrix

l|A|| =

On

the

other

)

the

condition

hand

(2.9)

operator

©£ =

setting

by

given

be

must

a bounded

if

and

only

in

-l

if

|

(2.10)

norm

air

ij

it

inverse,

an

has

Py

shi (2.8)

ee Di J

la,,|

.

for

uniform

continuity

of

(2.8)

is

that

sup Ilo, llSu coe s

(2750)

TY

Hence

can

we

find

shall

have

a sequence Efforts

is

Cy2)

led

to

the

Lorsany)

Q-matrix

(of

a counter-example

to

of Q-matrices

which

to

construct

following

9M =O;

\|exp

such

the

Markov

(2.10)

a sequence

group

holds having

conjecture

but so

if

we

Ilo, llSTEM.

far

failed,

one

conjecture.

there

whatever

for

order)

is

vanconsitanty

Ke—

K, (M4)

Such

that

every

finite

with

(-Q) || < M

(22)

satisfies

lgll O

with

at

such

least

that

one

every

zero

finite

diagonal

stochastic

element,

has

at

(of

least

one

eigen-

with

Rew

In

are

easily

there

ever

of

this

form,

=)

(9)

and

333

333 We

to

3.2%

in

deviations.

Then

normal

to

(1)

and

Xi,

can

be

A

K(A,B,C)

=E

DD

x)

and

and

this

= C5)

can

be

means

and

of

are

sometimes we

whilst.

which

to

ratio

bounds

is

6197/2"

what

(9)

the

upper

illustrate

(10)

of

1.0

(5)

independent

zero

for

ratio to

hand

To

be

with

written

Sy

lower

(Gly lO),

(9)

x,

the 0.5

= 0.636

consider

X50

0.5

other

worse.

than

distributions

(1)

the

the

tabulated

D =

mE sanCuecOiem

worse

Let

to

from

Thus On

sometimes

were

zero

and

unity.

OMS

return

(8)

mand mak(l, 1,0)

much

arguments.

distributed

to

and

is

to

from

in error.

Se O OSC

now

(10)

0.968103,

oI o Bool,

which

probabilistic

and

increased

(10)

oO. O57

(9)

again

about

(Coro) ee ()) ee

0.333

D

unity

increased

more

better

of

have

for

els

are

here

done

by

random

unit

equal.

variables

standard

as

Dy Bp

am ds} x,

ap 1)

a

Oeil

ee

X,

ae

(14)

ae Mae where

from

E

denotes

this

B? =l-

by

expectation.

simple

ce.

Then

It

probability (14)

can

be

D

=

15

er

Z

It

is

a

=

standard

a

BP Xx,

the

+

result

to

obtain

any

put

A=1,

However

inequalities

a

2

1 -

5 +

oo

pene nee aie of

a

[6 x

5

5

ox

BD

ar

Laney

distribution

the

eS Boke

Ee Rhee ras random

3 ae

:

aS

variable

Dene B x,

that

=1-B

written

as

Ae)

Consider

difficult

arguments.

=

(CEB

is

the

characteristic

funceLOnnOL

eZ)

sels

2

196

(ead

ane

Pee =

ycie Je a8)

er

hyx

re

n

i (210886)

a et

Oe

,:

es (a8

(16)

n=O

argument

than

greater

is

It

follows

of

polynomial

Legendre

the

is

P n (z)

where

order

its

that

Notice

_n.

:

unity.

that car

E[Z] = (208) nt P

(= :

Then

ratios

of

from

quadratic

theorem

forms

ese

.

a

2

2

in

+ B°x

Pitman

normal

A

X, + x,

this

in

(15)

we

-1

atis Cyr at

a

formula

used

obtained

(19) In

for as

the above

surface it

was

in

by

aoe

Pélya

rather

and

paper

area,

S(A,B,C),

shown

that

Szegd

2

that

Dien

egy


O

that, Sy “1

The

integral,

with

respect

°n2 s

-

to

(x, dx, ,2Z,)

N

l 5

=

Ane

“N

(8)

=

,

of

the

left-hand

side

of

(8)

over

all

N

(espre

1

hence

with

12>) so

€ E is

is

n,N

the

respect

therefore

left-hand

to

ure

greater’ than

side

,

Of

of

the

(6),

or

since

left-hand

equal

the

side

to

zy

2 nti,

latter

of

(8)

is

the

over

all

(E

and

nN

integral,

(X,Z)1Z5)

s?

N

It

follows

from

this

1 > m\

ye

nt+j

and

(7)

now

Proof

of

yee

Theorem

X

no (3), which

a

random

can

)

the

now

3.

First

n implies

Y

al | 01x

an

fix

> 0, n

:

diy

> in a

(x,

Oe

B,

i”

for

the

theorems.

fixed

B

«

martingale.

a

aril

=

[ohx8,

Qo

B,

the

This

(x,dv,z).

578 n X0b = Vi

property

the

definition

of

Zar

Q(x,

B,

z.) |
®™,

we

obtain

a!

lim

| | | |Q. (x,B,z,)

MA

sh

which

implies

Proof

of

the

the

to

1

IV aan

and

(10)

AL

Q”(x,B,Z,) | Ayay (dx

dz, dz.)

1}

there

B, n=

semigroup,

consistent

1)

is

ene

Mes

system

of

irreducible

two-sided

and

is

a semigroup

of

anda

probability

measure

such

that

for

t-almost

all

admits

i) (kp

am

on

8B,

n

Markov

Pp (x,B)

invariant

andall

TO 7,

lismamtri vial

probability

a stationary functions

xeS

CN)

lacy

conditional

transition

m

|

toliiowimgncoro

random exe

under

Be

ij, sim 23 WL,

of

Z,

Sp

the py =

ol

Gl)

yeB

in

the

the

sense

that

the

Radon-Nikodym

We view

of

density

remark

Theorem

1,

lim

left-hand

be

on

for

side

the

later

is,

as

a

right-hand

use

that

strengthened

| { lo. (x,;B,Z)

function

a

version

of

3 can,

in

side.

the

Corollary

to

Theorem

to

=

p"(x,B)

|1 (dx)

eee

(se.eha))

=

©

ss)

babrgeed

5

for

every

triangle

fixed

1S

m21.

inequality

This

can

be

proved

(x,

B)| $

by

induction,

since

implies 1

oe f
2m (A,)

(x_,x,) nlite? 17) and

«€ 5°

there

Peels

a)

-

1,

e

m(dx)

ts

eA

N

mes

cn eer

mms aD

N

Wiis) arn) are

intiy

1

not

=e sven z mutually

ee

N

A

p

NeaN

N

=

t(ax) AY

f

N

R SAO ies

eo

(x,A_)

N

=

(CoN

since Seo)

Be

Tr(Ae)

| 1(dx)

p

A

p :

Maz)

N

(x, dz)

|hen (dxki p=

i

+3 betetaeieet ie = 2n(a) - 1.

A

*

It

follows

x

€ AY

*

that

Beers

will

below.

singular.

See:

x,dz)

m(

U N=1

A.)

*

=

1.

Now

for

t-almost

all

x)

€

Ay

and

all

213

N

iy

x 1B)

jon

> |

=

|

z,€S

Z5€S

iy =.

|

2c

|

REG

she ates

nh

(X) /%5,B, 2) 12.)

i|

ots a lI

p

(x, ,dz,)

se]

p

(x,,dz,)

ail a8)

p

(x, ,dz,)

p

(x5,dz,)

3

Zz a.

in Since

the

same

lemma

will

be

is

Q

:

true

if we

|B ,) i.e.

since

i+j N

p

-N

-

p

sin

.

obvious

of

proved

trivralymeasure

is

(Xo,

show

B)

that

that.

(x) AY)

for

the

t

is

=) Op

MORSE’ The

version

[6]. 34

of

Om

ELor

assertion

Orey's

It follows,

j9

Corollary,

For

lim

m-almost

||p'(x,+)

€ AY

Lemma

convergence

therefore,

right-hand positive

*

XX

of

t-almost

i N +5 -N

iy

4

(15)

3.3

is

theorem

that

all

x O

Z,1Z

exactly

stated

from

Xo

when

(x, 1A)

and

all

A,

S.

the

is

a

non-

However,

this

and

€ Ay:

the

hypothesis

proved

as

ina

Theorem

2.1

in

that

= 0

noo

4.

Uniqueness In

the

Let

Markov

random

{Q",m 2a n §3 for the second

fields

first

process,

O8e

tion

functions

—- ©

respectively.

4.1

Lemma.

B

associated

With

The the

and

p'(-,°)

are and

probability notation

of

in

prove

cee

X

Theorem

eK

Kp

be

two

stationary

irreducible system of kernels n jo) (Geie)) 52 (ip 2 e Spi hy) aS

p(x,B)

Markov

in

a similar

random

fields

fact

stationary

Markov

pr(+,°)

and

measures 82,

2.

eee

the

bevel

m(B),

1 the

shall

and

with

1B )

Theorem

< i < o}

we

Xp ece

process, By

qT,

Be

section

10K

nea) ae Deriners:

and

Proof.

present soeek

for

mt

stationary

and

m-almost

7

are all

{X5,

way = >)

processes

for 2

the al & ca}

with

distributions

not Xx) €

mutually Serand

: alin

transitm

and

singular.

sala

Xo e

Sy

214

al p Se)

= |

iN

f

Q, (x, By,2)

(X) 1X5 /BrZ) 1Zy)

p

BL NaN ral

p

(a

2

dzan

1,73 Nw@eN

(x,,dz,)

(x

3 Integrating

=

sides

with

iy ||ae (x) 1X5 1B, 2112)

ss

ss yt

y

applies

to

7 (dx,)p

Since

or

respect

|

=

mee)

both

the

equal

same

to

the

SVAN

(x

2

m (dx, ) 7 (dx,)

m (dx, ) p

iytiy

over

side

| |™ (dx, ) p

we

obtain

(x) dz, )

(16)

it

of

follows

that

(mAm)(B)

sufficiently

(16).

If

N

is

> O

and

similarly

is

1yt3y implies

:

S

z

m™(B),

right-hand

to

(x, ,dz,)

large,

~

m

for

greater

than

(14)

yn

p

and

A_A N N

Since hand

ivan) x N(x 1X ASinrds 1°22 a ae,

side

Hence

of

(16)

(mAt)(S)

defines

> 0,

Now,

there

is

a

T (dx)

p- -

Be

to

B

2

M@

establish

that

the

in

B,

which

is

positive

the

right-

for

B=S.

lemma.

Theorem

7(B)

SX le

CYL

a measure

proving

such

attoae

#

2 we

show

first

that

7(B).

For

fixed

m

7

> 1,

=

T.

the

Suppose

measure

+

by

(x,dz)

Corollary

3.4.

lim n>

Now

for

each

converges

Thais

in

total

(13)!

imply

|| Q” (x,B,z) Ss

i =

and’

variation

to

- p"(x,B) | m(dx)

m(dx)

waz)

1,;255..

(choose

first

an

—

m,

such

that

1

itl

iS)

by

Corollary

=0.

*

aa {lp (,B) = (B) | a(dx)
2(l+r)/(1-r),

As

that

of

assumptions

2.

modulus

the

that

(C(A),

{X,:

{Z (A): A « A}

minimal

finite

for

+

nv

s_(A) > Q

necessitates

study

random

and under

Brownian

=

|

1°)

of

distributed

Ol1ny

restriction

that

upon

makes

k

n

KO

AS)

a positive

Lindl dete

integer.

e( 2

nlornthic

229 t

F

sup{|2*

(A\A,

)|: AeA}

-k/2

OM.

of

(2.16)

and

Substitution

divergence

(2.17)

(2.18)

In(t

k k n'B ”) = 1n A, + (1/2 + 1/s)k Inn (le/

= oe

©

into

the

a

-

(2.18)

allow

c= ive

to

KO

For

this that

to

happen

the

2a) /S))keeln

s

Since

this

We) Senesuay

>

ko

bound

and 6,n

ea

oance

an

to

-k

£C/2

=

+ Yat

Ink

(r+b)

choose

Inn

(ep)

-

l/s)k

linn

(iewr so \/s her sb ne + (1+ (r4b)

is

Vine

Fund

ias

n

Ink. =o(ln positive;

n)

that

and

to

choose

s

is,

(279)

increasing

function

Absa)

(iets)

of

b

we

can

always

find

¢

b

> 0

(ZO)

inequalityis the

assumption

It

show

and

must

n

tae

remains

ieee

meters Of

is

(asks)

latter

ao

suffices of

y

2(1+rt+b)/(1-r-b).

SS

This

it

and

n

+ (1n B)k,

ne

tn my Pde /2 =

coefficient

that

of

+

so

shows

it y

now

is so

to

possible that

(2.3),

to

that

made for

choose (2.4)

in

the

Theorem above

values

and

(2.5)

for

1.

choices

the

hold

of

8,

remaining

for

any

b,

s,

para-

preassigned

230

eet a choice

of

y

"5 < nee et sufficiently

Expression

that

the

(2.16)

event

in

truncated

that

never

(2.7)

exponent

its

o(1)

value

¥

=rj

B

indicates for

Aigad Boe

be

the

small

will by

insure

n

for

probability

made

that

construction

large

the

bound

arbitrarily

(2.11)

when

+b)

Nae

aul

that

which

lap thatern

as

value

a

j

holds. (2.18)

truncated

equal

insures

processes,

truncated

small.

To

check

written

in

the

and

(2.3),

form

while

nonobserve

(2.12)

is

5

(Dean )

iL

increases,

then

converges

ee ae

(2.5) since

the of

a ee TeCT EY ae Sa =

here

J=k oe

= A ee

holds

for

that

can

of

- 2K6 7

where

occurs

insures

processes

the

(2.4)

ye

the to

decreases

to

term

zero its

decreases

as

n+

value

to

by

for

(2.13).

Irk

which

-

in

turn

converges

thetical

6.

term

to

in

Consequently

these

sums

small,

can

since

multiplier

the

is

Since

not

for

sum be

only

Te

last Vltew

the

case

w(Z

5)

(2.3)

remains

in

arbitrarily of

6

we

choose

bounded

bounded small

after

by

our

y=6" /8Kn,

away

from

as

n-+

©.

choosing

choice

of

the

zero

The

46 y

paren-

uniformly

in

bound

on

sufficiently is

as

the

front

use

derivations the

s w(Z


Ti

ergodicity

and

occurs

these

are

and

to

jofsonden

depend

on

are

A;

and

convergence

cases

atom

{x}

convergence

criteria

convergence

(253)

P"(A,A)

local

proofs

deriving

to

3 ig

contains

an

SO

ergodicity

coupling

S

pointwise

1s) Stiilelacking

ss ej

exists

[18]

(Ze)

countable

point

some

Section

than

above.

that

is

every

for

geometric

between

leemset there

be

S

stronger

required

in

require

suppose

(When

if

x

and

as

only

natural

then

property,

case

are

criteria

to

as

result,

wy

limits

Wt

[25]

(such

slightly

the

example,

all

are

forms

sense,

Harris

that

(2.3)

satisfactory,

> 0O.

x

the

see

For

(A)

similar

general

relationships

> O

contexts

all

arguments the

timeseucal o(A)

(a)

Ny

°

of order

and

where

more

usual

A

the

We

on

Yo €

0,0n Se,

(2.2)

shall

m-almost

rate.

of

[17],

with

Sscoludarity

proofs

able

exists

y(n)

yw=O

ergodic

- all>

everywhere.

nt—-almost

geometric

Global

oe

{x}

We and

ysome

a point

holds

and

set.

stronger,

only

The

there

are

definitions

[18],

a

in

a

4} Se,

se E€ So

The

outside

aul

n)® exp (yn°)

pio) IP n (,°) OMe

seore

n-> Examples

=

at a

Wr.

generating

not

avail-

used. for

(2.1)-(2.3)

moments

fomvevyeny,

Vale

of

F

based

return

wath

264

conditions

relatively

compact

set

of

convergence

of

rates

1:

Theorem

(i)

Our

moments

and

If

B

x first

is

result

small.

set

B

known

probability

and

expectation

summarises

iow (Goh 22 Ales

ae

by

relations

between

:

of

small

every

P(x, 7,

also

for

respectively

BP

and-

Sand sweeter Xe) B)g) n x conditional on Xo =x.

a

on

time

hitting

the

is

$-measure

positive

general

fairly

under

probabilities

transition

the

on

denote

us

Let

[16]

in

whilst

o¢-irreducibility; continuity

of

a consequence

as that

out

pointed

is

it

small,

is

atoms

of

collection

finite

every

that

Note

< «0,

sup E, (t,)

and

and

ae

=]

eco;

xeB 'B

ne

and

)) Sas

Gai)

6

where.

geometrically Wie,

some

sup

known

well

is

splitting

Of

3(ii%)

yy,

Gxeae Bull

at

Ee al eo

B

if

technique

of

is Theorem

ergodic.

E,(y

O

SES),

Ca

—telaveia

is

{x}

yw

Nummelin's

Theorem

and

sa)

order

(i)

Gi)

is

{x} for

and:

xeB

ie

of

Proof:

assmalie

n

Wl (n=

ergodic

Be

eleteya

a

Viele GULL ste

[16];

4(2)

and

Converses

and

(12)

to

is

4(21)) all

Proposition

Theorem

of

these

and

atom,

an

is

3 of

3(42)

[25],

and

as

3(aii)

from

general

in

follows

in

of

the

proof

L167

and

Fl7)-

0

results

hold,

and

for

completeness

we

give

Theonem,

22)

(a)

i 1 (dy) ey (pice)

Ne

if,

anu

set

jigs

O

Proof: andi

AY eh;

Ce

(ini)

{x}

is

{x}

is

Harris

ergodic,

then

for

any

Ae

F,

we = i

ase

{x}

geometrically

Suel

is ergodic

fe mite)

pie

ergodic, then for T i m (dy) BE, (x A) < »,

isaeic

of order

de) < ol,

andi

, fom

with

we

T—almosity

any

set

A,

then

alll

for

3x eS!

ees

(i)

is

classical

ais) Theorems) In

in

See

Exealeies

(GERD)

EL (y

ee

= Ge

particular,

ff

[16] the

(see

OFM

and

[17]

range

of

Aas)

for

example

[3]); (ii)

is Theorem

3(i)

of

[16];

7ie more

detailed

measures

cay) [leo Ge,

results

\

and

ee

n

wu

(ys)!

are for

also

which

given the

concerning,

quantities

265

have

the

what

we

appropriate We

convergence the

shall

believe

to

and

analogues

3.

will

not

these

give

the

hitting

of

most

time

Foster's

Criteria

be

behaviour.

be

Suppose

g

limiting

a test

details

important

moments,

criterion

for

finiteness

that

g(x)

is

for

the

function

we for

of a

here.

Rather,

connections

shall

in

the

these

various

hitting

time

non-negative

various

having

between

next

section

moments

to

of

describe

be

finite.

moments

measurable

types

summarised

rates

of

function

ergodicity

on

above.

e¢

TeoreniSG)e

arp

for

J, P(xrdy)

some:

gly)

|e >10),

< glx)

“and some

- ,

Ae

fF,

c

xe A,

er

then

E

(t,)

(Sal)

=. oiled) Sy

ne

@(Gs)

se EIN

Sil

ay mg Gy) te Pico

a

ee.

oe a IN,

Glee!

ele

ocx), 8 exare

c

5

(3

Store Oe

= ©

(ee

ru

then

, = sa(las))ip

ca

YS

E(x

for

any

a

ees

Se

Ek

a

(3

GS)

66

(Grin

gC

ihe Dice,chy)

SE (elt, )) 7for

cis)