Random Fourier Series with Applications to Harmonic Analysis. (AM-101), Volume 101 9781400881536

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Annals of Mathematics Studies Number 101

RANDOM FOURIER SERIES WITH APPLICATIONS TO HARMONIC ANALYSIS BY

MICHAEL B. MARCUS AND

GILLES PISIER

PRINCETON UNIVERSITY PRESS AND UNIVERSITY OF TOKYO PRESS PRINCETON, NEW JERSEY 1981

Copyright © 1981 by Princeton University Press ALL RIGHTS RESERVED

The publishers are grateful for the assistance o f the Andrew W. M ellon Foundation in the publication o f this book

Published in Japan exclusively by University o f Tokyo Press; In other parts of the world by Princeton University Press

Printed in the U nited States o f Am erica by Princeton University Press, Princeton, N ew Jersey

Library of Congress Cataloging in Publication data will be found on the last printed page o f this book

CONTENTS

C H A P T E R I:

IN T R O D U C T IO N

C H A P T E R II:

3

P R E L IM IN A R IE S

1. Covering number of compact metric spaces 2. A Jensen type inequality for the non-decreasing rearrangement of non-negative stochastic processes 3. Continuity of Gaussian and sub-Gaussian processes 4. Sums of Banach space valued random variables

16 19 24 40

C H A P T E R III: RANDOM F O U R IE R SERIES ON L O C A L L Y C O M P A C T A B E L IA N G R O U PS 1. 2. 3.

Continuity of random Fourier series Random Fourier series on the real line Random Fourier series on compact Abelian groups

C H A P T E R IV: TH E C E N T R A L LIM IT TH EOREM AND R E L A T E D Q U E STIO N S

51 60 63

65

C H A P T E R V: RANDOM F O U R IE R SERIES ON C O M PA C T N O N -A B E L IA N G R O U PS 1. 2. 3.

Introduction Random series with coefficients in a Banach space Continuity of random Fourier series

C H A P T E R VI: 1. 2.

1. 2. 3. 4.

A P P L IC A T IO N S TO HARMONIC A N A L Y S IS 105 118

The Duality Theorem Applications to Sidon sets

C H A P T E R VII:

74 81 93

A D D IT IO N A L R E S U L T S A N D COMMENTS

Derivation of classical results Almost sure almost periodicity On left and right almost sure continuity Generalizations

122 134 138 140

REFERENCES

144

IN D EX OF T E R M IN O L O G Y

148

IN D EX OF N O T A T IO N S

149

v

Random Fourier Series With Applications to Harmonic Analysis

CHAPTER I IN T R O D U C T IO N

A b s t r a c t : N e c e s s a r y and sufficient conditions are obtained for the a.s. uniform convergence of random Fou rier ser ies on locally compact A b e li a n groups and on compact non A b e li a n groups. are obtained.

Many related results such as a central limit theorem

The methods develo pe d are used to study questions in harmonic

a n aly s is which are not intrinsically random.

In a series of three papers published in 1930 and 1931, Paley and Zygmund [44] studied a variety of problems concerning series of indepen­ dent random functions and raised the question of the uniform convergence a.s. of the random Fourier series CO

(1.1)

^

c nEnein X ’

x (

[0,277] ,

n=0

OO where

ic„S are real n

numbers,

>n c « =

1 ,n and

!e„)is a Rademacher

n= 0

sequence, i.e. a sequence of independent random variables each one taking the value plus and minus one with equal probability. considered (1.1) with where uted on

They also

\zn \ replaced by a Steinhaus sequence

le

12770)

i,

i 0

n=2

but not necessarily if

e =0 . A lso they introduced the numbers

3

4

R A N D O M F O U R IE R S E R IE S

2j+1“ 1 S: =

.V4

0 • We have the following version of

k->oo Theorem 1.1 in this case. TH EOR EM 1.3.

Let

||Q|| =

sup

|Q(t)|

and let o be defined as in

tf [o.l]

(1.12)

then if 1(a) < oo the series (1.17) converges uniformly a.s. and

2

00 (1.18)

(E||Q||2) l/2 < C '(s u p E P k )V2[ ( S

where C '

is a constant independent of

open sets

U c [0 , 1 ]

V

ak ) * +

^

= 00 ^ en ^or

11

IN T R O D U C T IO N

(1.19)

sup sup | V n

a kekpkcos (Akt + $ k)| = oo a.s.

t J f i , P ~ mg(e)

(1.3)

Proof.

Let

B(t, e) = lx p {B (0 ,e ) fl K © K i 4

and (1.5)

M ^ (K © K ,e/ 2 ) > N p (K © K ,e ) .

Inequality (1.4) is elementary since for t e K © K , {B (t,

s)H©KS.

t + iB (0 , e) H K © K| C

Inequality (1.5) is a well-known fact which is proved as

4

follow s:

Denote the centers of the

M ^ (K © K ,e/ 2 ) b alls by

11j; j = 1, — ,

M p(K © K , e/2)I. (T h ese balls are not unique and if p is a pseudo-metric neither are the tj

but that d o esn ’t affect the proof.) We have Mp(K©K, e/2)

(1-6)

(J

K®K C

B (t j, e) .

j= l To see this suppose (1.6) is false.

Then there exists an s e K © K

but

Mp(K©K, e/2)

(J

not contained in

B (t j,e ).

For such an s

we have p (s ,t j)> g

j= l for a ll

tj . Let

u e B (tj, e/2) then

p (s ,u ) > p(s, tj) - p (t j,u ) > e/2 .

Therefore

B (s ,e / 2 )

is disjoint from B (tj,e / 2 )

for a ll

j , 1 < j < M ^(K © K ,

e/2) and this contradicts the assumption that M ^(K © K , e/2) is maximal. Thus we have established (1.6) and (1.5) follow s immediately. U sin g (1.4) and (1.5) we see that Mp(K©K, e/2)

> /*(

U

lB (t j,E / 2 )n ® K l)

> Mp (K ® K, e/2) /i(B(0, e/2) fl K ® K ) > Np (K ® K ,e )m g (e/2) which is (1.2).

18

R A N D O M F O U R IE R S E R IE S

To prove (1.3) we note that, analogous to (1.4), for a ll (1.7)

t cK

/ z{B (0 ,e)riK © K | > /x{B(t,6)nK | .

Now suppose we have a minimal cover of

K by balls of radius

respect to p with centers in K . There are we denote their centers by

N (K,

e

)

e with

balls in this cover;

itj , l < j < N (K, e)! . Then, since

N p (K , e)

KC

[J

B (t -,g ), J'

we have

j= l N

H=

(K , e)

U BnKl j= i

< Np(K, €>/iiB(0, E )flK ® K i

where we use (1.7) at the last step. R E M A R K 1.2.

1.1

Thus we obtain (1.3).

In the special case that G

is compact and

K = G

Lemma

is just

(1 8 )

where

mg(s) -

NPp (G ’ £) £ ^ rrm mg(£/2)

m^(s) = p (x eG \ 8 (x ) < e) . This result is elementary and easy to

prove directly. In the next lemma we relate the covering number of covering number of L E M M A 1.3.

Using the notation given above Np (K ,2 e) < N p (K © K ,E );

(1.10)

Np (K ® K ,2 £)
2a for a ll

p (t j,u ) > p (s j , u ) - p ( s j , tj) > e, cover

For each non-empty set choose an

Sj . But then

contradicting the fact that the

lB (tj,e )i

K.

We now prove (1.10). with centers

Let

B (t j,e ),

1 < j < N (K, a) be a cover for

K

tj e K . It follow s that N

(K ,e )

(J

{B (t i ( 6 )® B (tj,6 )l

i,j= l

is a cover for u- € B (t-, e)

K®K.

Let

s 6 B (t-, e )© B (tj, a ) .

Then s = u - + V j

for some

and Vj e B (t j, e) . We have

p (s , t^ -f tj) < p {ui , t i) + p (vj , tj) < 2a .

Therefore

{B (t-, e )® B (tj, e)i C B(t^ + tj, 2a) and consequently N p (K , a)

U

B (tj + t j, 2e)

ij= l

is a cover for

K®K,

with centers in

K®K,

consisting of

N ^(K , a) sets.

This gives (1.10). 2.

A Jensen type inequality for the non-decreasing rearrangement of non­ negative stochastic processes Let

C

and assume

be a compact subset of a locally compact A belian group that G

is not discrete.

Let

g ( x ) , x e C be a real valued

non-negative measurable function on C . Define /xg(e) = /z(x fC | g (x ) < e)

where, as above,

G

p denotes the Haar measure of G . Set

20

R A N D O M F O U R IE R S E R IE S

g(u) = supiy |/Xg(y) < ui ;

g

is called the non-decreasing rearrangement of

Since that

0 < //^ < //(C )

the domain of g

g (with respect to C ).

is the interval

g viewed as a random variable on [0, //(C )]

[0, /Lt(C)]. We note

has the same probability

distribution with respect to normalized Lesbesgu e measure on

[0, //(C)]

that g has with respect to normalized Haar measure on C . In particular

(2 . 1)

Lemma 2.1, which is a generalization of a well-known observation, pro­ vides an alternate definition of g . Lem m a 2.1.

For 0 < h < //(C)

(2 .2) 0

//(E)=h

E

i.e. the infimum is taken o v e r a ll // measurable subsets

E

of C

for

which //(E) = h . Also, for D a non-negative constant we have

(2.3)

Proof.

Let

E

be a measurable subset of C , //(E) = h . Let

fi\xeE | g (x )< e i .

Then

//(e) < /zg(e) and

where the last step is a statement of (2.1).

//(e) =

P R E L IM IN A R IE S

21

We complete the proof of (2.2) by exhibiting a set F ) = h and for which equality is attained in (2.2).

F C C

such that

This is quite simple

when Alu e [0, h]|g(h) = g(u)S = 0 , where A denotes Lesbesgu e measure. We consider this case first.

Since

g and

g are equally distributed, as

random variables, on their respective probability spaces, the probability distribution of

g ( u ) , for u e [0, h) with respect to normalized Lesbesgu e

measure is the same as the probability distribution of

g (x ),

g (x ) < g(h)i

This implies that

with respect to normalized Haar measure.

for x e i0
0 .

We can alw ays decompose the set

into two disjoint sets

F1 and

F2 with /x(Fx) = Sj .

Therefore we can write

g(u )d u =

J 0

J*

g(x)/z(dx) .

lo < g (x )< i(h )iU F 1

The domain of integration of the integral on the right is the set completes the proof of (2.2).

(2 .2).

F . This

The equality (2.3) follow s immediately from

R A N D O M F O U R IE R S E R IE S

22

The next lemma is also a generalization of a well-known observation. It displays the property of the non-decreasing rearrangement which lies behind Lemma 2.3. L E M M A 2.2.

C .

Its proof is immediate from Lemma 2.1. g(x ) and f(x )

Let

be non-negative measurable functions on

Then for 0 < h < fi(C) h

j

(2.5)

h g+f(x)/z(dx)

> J

°0

h

0

0

We now present the main result of this section. probability space and let

||

||q

Let

(P ),

cue A ,

then

(A ,? , P )

be a

be a norm on the linear space of real

valued functions on (fl,? , P ) with the property that if a.s.

.

g ( x ) ^ ( d x ) + f 7(x)/e(dx)

||Y||q > ||X ||q . Let

g(x,cu),

|Y(cu)| > |X(co)| xeC

be a non­

negative measurable function on C x fl . The following lemma generalizes Lemma 1.1 [36]. L E M M A 2.3.

llg(x > -)IIq
0

be a non-increasing function for

0 < u < fi(C ) and set fi = fi(C ) . Then

(2.6)

||J '

g(u, w)f(u)du||n
^

^ - f1 ’

h (2.7)

II j

h g(u, cu)du ||^2 < J

^0 Proof.

||g(u, *)I!q f(u )d u .

f

Ilg(u, O 'l^du .

0

We w ill first obtain (2.7).

By Lemma 2.1, for each

h

J

g(u, cu)du < 0

J* g(x, co) fi(dx) E

co e (fl,? , P ) ,

23

P R E L IM IN A R IE S

for a ll

E C C

with /x(E) = h . Therefore h II j

g(u, HqMdx) •

E

E

By Lemma 2.1, the right side of (2.8) is equal to the right-hand side of (2.7). There is nothing to prove in (2.6) unless the right side is finite. this case the integral on the left in (2.6) is finite on a set P (Q ) = 1 . This implies, since

||

||q

f

II lim g(v,&))dvf(u)||jj = 0 . u- ° J 0

The finiteness of the right side of (2.6) also implies that

lim

(2.10)

f

||g(v,.)HQ d vf(u ) = 0 .

u- ° J o Integrating by parts and using (2.9) and (2.10) we have

(2.11)

II f

g (u , 0 ))f(u)du||Q < II f

Jo

J

g (u , 0 ))du||n % )

q

►H- />u u + II I

n

I

e(v,co)dv d(-f(u))||Q .

.

Q C Q ,

is (by assumption) a lattice norm,

that (2.9)

In

24

R A N D O M F O U R IE R S E R IE S

U sing (2.7) and another integration by parts we have

n

u

_______

g (v ,c j)d v d(-f(u))||Q
2

oo (3.7)

Pi

U

oo V

n=no We w ill choose For any

that

Pi

[J

2

g (n) = G (n Q) .

n=n0

ib n S below such that

s eT

lim r ( s , s n) = 0. n->oo oo


0 there exists an n^ such

oo (J

nQ^ oo.

A n and a ll

nQ > n^

27

P R E L IM IN A R IE S

sup

! Y (s n ) c u ) - Y ( s m,w )l


nn

For these

co we define

sequence

Y (s

Consider

n =n ^

Y(s,cl>) as the limit of the appropriate Cauchy

,co). s, t c T

such that r(s, t) < 2

^ ft , nQ > n^ . We w ill show

oo

that if co /

|J n=n

A n , then q u

oo

(3.8)

lY (s ,c u )-Y (t,< o )! < 3

^

bfl .

n=n0

This shows that the function

Y (s,cu ) just defined is uniformly continuous

on T . Since this can be done for a ll separable version

iY (t),t< rT i

> 0 we obtain a continuous

e

of iY (t),t< rT i

and, as a trivial consequence,

S of S. To obtain (3.8) consider that for a ll

-n 0 such that r(s, t) < 2 . A lso note

s, t c T

n > nQ there exist

and r(t, tR) < 2~n . Hence r(s

s n and ,t

tn e A n such that r (s ,s n) < 2 n

) < 2

°

, i ( s n, s n+1) < 2“ n+1

OO r (tn, in + i) < 2_n+1 . Therefore, if

(3 .9 )

cu i

(J A n n=n Q

l Y ( s , w ) - Y ( t , w ) l < !Y (s n0 oo

,c u )- Y (t n ,co) ! r0

oo

+|Y(tn, c o )- Y (t n+1,w)|

+

n=n0

]£

|Y(sn, < u ) - Y ( s n+1, w)|

n=nQ

from which we get (3.8). We also have, by (3.7) and (3.8) oo

(3.10)

P {

sup_ r(s,t)< 2

| Y (s )-Y (t)| n°

> 3 ^ n=n0

b n } < G (n 0) .

and

28

R A N D O M F O U R IE R S E R IE S

For X > 1 we set

bn = ^ = 8

[1° g Nr (T >2_n

) +1° g "1

then

C3.li)

i

3. < « { i

♦ i

on-t-i

_n = n 0

0n

n z = n o

1

J ^O " 1

< A 25

2 I L

Substituting

(log Nr (T , u ))1/2du + —i - -

(3

2 0

b 2 into G (n Q) we get

oo (3.12)

(log n0) ‘/2

G (n 0) = 8

r

^

N 2(T , 2~n)e x p [-8 A 2(log N r(T , 2~n~ x)+ l o g n)]

n= n 0

< 8 exp [-6 log N r(T, 2

n 1 0 )]

°° exp [-8 A 2 log n] .

The inequalities (3.11) and (3.12) prove our assertions about

£ b n and

G (n 0). Let

1

Z=

sup r(s,t) 0 but for

our purposes we need only note that oo EZ2