Probabilistic Applications of Tauberian Theorems 9783110195293

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Probabilistic Applications of Tauberian Theorems

A L S O AVAILABLE IN M O D E R N PROBABILITY AND STATISTICS: Queueing Theory P.P. Bocharov, C. D'Apice, Α. V. Pechinkirt and S. Salerno Generalized Poisson Models and their Applications in Insurance and Finance V.E. Bening and V.Yu. Korolev Robustness in Data Analysis: criteria and methods G.L. Shevlyakov and N.O. Vilchevski Asymptotic Theory of Testing Statistical Hypotheses: Efficient Statistics, Optimality, Power Loss and Deficiency V.E. Bening Selected Topics in Characteristic Functions N.G. Ushakov Chance and Stability. Stable Distributions and their Applcations V.M. Zolotarevand V.V. Uchaikin Normal Approximation: New Results, Methods and Problems V.V. Senatov Modern Theory of Summation of Random Variables V.M. Zolotarev

Probabilistic Applications of Tauberian Theorems A.L. Yakimiv Translated by Andrei V.Kolchin

///VSP///

ISBN: 90 6764 437 4 © Copyright 2005 by Koninklijke Brill NV, Leiden, The Netherlands. Koninklijke Brill NV incorporates the imprints Brill Academic Publishers, Martinus Nij hoff Publishers and VSP A C.I.P. record for this book is available from the Library of Congress All rights reserved. No part of this publication may be reproduced, translated, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission from the publisher. Authorization to photocopy items for internal or personal use is granted by Brill provided that the appropriate fees are paid directly to the Copyright Clearance Center, 222 Rosewood Drive, Suite 910, Danvers, MA 01923, USA. Fees are subject to change.

Contents Introduction

vii

1

Tauberian theorems 1.1 Regularly varying functions in a cone 1.2 Weak convergence of measures and functions 1.3 Multidimensional Tauberian theorems of Karamata type 1.4 Weakly oscillating functions 1.5 A multidimensional Tauberian comparison theorem 1.6 One-dimensional Tauberian theorems 1.7 Tauberian theorems of Drozhzhinov-Zavyalov type 1.8 Three multidimensional Tauberian theorems 1.9 Remarks to Chapter 1

1 1 9 16 26 38 51 70 76 86

2

Applications to branching processes 2.1 Bounded below branching processes 2.2 Bellman-Harris branching processes 2.3 Convergence of finite-dimensional distributions 2.4 The number of long-living particles

89 89 93 98 106

3

Random ^-permutations 3.1 The number of/1-permutations 3.2 Auxiliary limit theorems 3.3 Fundamental limit theorems 3.4 Uniformly distributed sequences 3.5 Examples of sets A 3.6 Random sets A

111 Ill 117 126 137 142 158

4

Infinitely divisible distributions 4.1 Probabilities of large deviations 4.2 Asymptotic behaviour of a density at infinity 4.3 Multidimensional case

167 167 181 192

5

Limit theorems in the record model 5.1 Intervals between state change times in the record process

197 197

5.2

206

The asymptotic behaviour of the kth record times

Bibliography

211

Index

225 ν

vi

Introduction By Abelian theorems are meant those assertions which allow to deduce from the asymptotic behaviour of sequences and functions the asymptotic properties of their generating functions and Laplace transforms (as well as other integral transforms). Theorems converse to Abelian are referred to as Tauberian. They are named after Abel and Tauber, respectively, who were the first to prove theorems of such kinds (Abel, 1826; Tauber, 1897)). Usually, direct methods are used to prove Abelian theorems. It is much more difficult to prove the corresponding Tauberian theorems, and a wide spectrum of analytical techniques is involved. As milestones in Tauberian theory, we mention the works (Littlewood, 1910; Hardy, Littlewood, 1914; Karamata, 1930b; Karamata, 1931a; Wiener, 1932; Korevaar, 1954; Keldysh, 1973; Vladimirov, 1978). In last three decades, much thought has been given to multidimensional Tauberian theory. This is primarily due to the fact that Tauberian theorems are finding ever-widening application in mathematical physics, theory of differential equations, and probability theory. We place particular emphasis on the multidimensional studies (Alpar, 1976; Alpar, 1984; Vladimirov, 1978; Vladimirov et al., 1988; Drozhzhinov, 1983; Drozhzhinov, Zavyalov, 1984; Drozhzhinov, Zavyalov, 1986a; Drozhzhinov, Zavyalov, 1986b; Drozhzhinov, Zavyalov, 1990; Drozhzhinov, Zavyalov, 1992; Drozhzhinov, Zavyalov, 1995a; Drozhzhinov, Zavyalov, 1995b; Drozhzhinov, Zavyalov, 1998; Drozhzhinov, Zavyalov, 2000; Drozhzhinov, Zavyalov, 2002; de Haan, Omey, 1983; de Haan et al., 1984; Resnick, 1991; Omey, 1989; Omey, Willekens, 1989; Diamond, 1987; Kozlov, 1983; Pilipovic et al., 1990; Stankovic, 1983; Stankovic, 1985a; Stankovic, 1985b; Stadtmüller, Trautner, 1979; Stadtmüller, Trautner, 1981; Stadtmüller, 1983; Stam, 1977; Chelidze, 1977). We omit the discussion of Tauberian theorems with remainder term; the reader can find the details in the books (Aljanöic et al., 1974; Ganelius, 1971; Postnikov, 1980; Subkhankulov, 1966) and the papers (Frennemo, 1965; Frennemo, 1966; Vladimirov et al., 1988; Drozhzhinov, Zavyalov, 1995a; Drozhzhinov, Zavyalov, 1995b). We note, though, that the use of ordinary Tauberian theorems (without a remainder term) has allowed the author to obtain an exact asymptotic expression of the remainder term for infinitely divisible distributions (see Chapter 4). The Tauberian theory has found a widespread application in probability theory. Tauberian theorems have been used to study asymptotic problems of probability theory by (Vatutin, 1977b; Vatutin, 1977c; Vatutin, 1979; Vatutin, Sagitov, 1988a; Zolotarev, 1961; Novikov, 1982; Postnikov, 1980; Rogozin, 2002a; Rogozin, 2002b; Sevastyanov, 1978; Seneta, 1969; Seneta, 1973; Seneta, 1974; Feller, 1966; Bingham, 1984a; Bingham, 1984b; vii

Bingham, 1988; Bingham, 1989; Bingham, Doney, 1974; de Haan, Omey, 1983; de Haan et al„ 1984; Omey, 1989; Weiner, 1990)), to name a few. Despite of the strong interest of probabilists to Tauberian theorems, no book specially devoted to this topic has been published yet. This monograph is intended to fill this gap. A series of studies (Yakymiv, 1981; Yakymiv, 1982; Yakymiv, 1983; Yakymiv, 1984; Yakymiv, 1987a; Yakymiv, 1987b; Yakymiv, 1988; Yakymiv, 1990a; Yakymiv, 1990b; Yakymiv, 1991a; Yakymiv, 1991b; Yakymiv, 1993a; Yakymiv, 1993b; Yakymiv, 1995; Yakymiv, 1997; Yakymiv, 2000; Yakymiv, 2001; Yakymiv, 2002; Yakymiv, 2002; Yakymiv, 2003a; Yakymiv, 2003b), which have seen the light in the last two decades, are also devoted to probabilistic applications of Tauberian theorems and form the basis for this book. It contains Tauberian theorems and their applications to analysing the asymptotic behaviour of stochastic processes, record processes, random permutations, and infinitely divisible random variables. We include the works on branching processes (Vatutin, 1977b; Sevastyanov, 1978) which give the impetus to our studies in this field. We also include a series of Tauberian theorems due to Drozhzhinov and Zavyalov which, we believe, are of much interest to probabilists, although they were intended for use in other fields of mathematics. The book on Tauberian theorems and their applications follows the traditions of the Steklov Institute of Mathematics. It suffices to mention the books (Postnikov, 1988; Vladimirov et al., 1988) and the papers (Keldysh, 1973; Sevastyanov, 1978; Vatutin, 1977b). Tauberian theorems are contained in the first chapter of the book. In particular, multidimensional extensions of Tauberian theorems due to Karamata are given in Section 1.3; a multidimensional Tauberian comparison theorem of Keldysh type and an extension of a Tauberian theorem of Littlewood type are given in Section 1.5; Section 1.6 contains a series of one-dimensional Tauberian theorems. The whole Section 1.7 is devoted to Tauberian theorems due to Drozhzhinov and Zavyalov. In Section 1.8, three multidimensional Tauberian theorems of Drozhzhinov-Zavyalov type are given, which are used later to study the asymptotic behaviour of infinitely divisible distributions in a cone. Sections 1.1,1.2 and 1.4 are of auxiliary nature; here we deal with generalisations of regularly varying functions occurring in Tauberian theorems. Chapters 2-5 cover probabilistic applications of Tauberian theorems. In Chapter 2, asymptotic properties of branching processes are studied. A series of limit theorems on random permutations whose cycle lengths belong to a given set A (the so-called ^-permutations) constitute Chapter 3. Chapter 4 is devoted to analysing the asymptotic behaviour of infinitely divisible distributions at infinity. In Chapter 5, probabilities of large deviations for some random variables are studied in the context of the record model. The author is sincerely grateful to his colleagues of the Steklov Institute of Mathematics, Professors V. A. Vatutin, Yu. N. Drozhzhinov, A. M. Zubkov, V. F. Kolchin, A. I. Pavlov, V. E. Tarakanov, to Professor V. Yu. Korolev of the Moscow State University for their corrections and remarks, and to Professor Α. V. Kolchin who kindly agreed to translate the book to English.

1

Tauberian theorems 1.1.

Regularly varying functions in a cone

A Borel set Γ c R" is said to be a cone with apex at a point a € R" if a + λ(χ — a) e Γ for any χ e Γ, λ > 0, λ e R . Let int A denote the interiority of the set A c R". We say that a cone Γ is solid if int Γ φ 0 . A closed solid convex cone is said to be acute if there exists a hyperplane of dimensionality η - 1 which meets the cone at its apex only. Let Γ be a cone in R" with apex at zero, S = Γ \ {0}. We fix some vector e € S. Everywhere in this chapter / is a real positive variable. A function / ( x ) is said to be regularly varying at infinity along Γ if it is defined for all x € S, |x| > c > 0, is positive and measurable on this set, and / ( f * ) / / ( f e ) - > * > ( * ) > 0,

ψ(χ) 0 and some real p. Furthermore, for any χ € S and λ > 0 by (1.1.1) as t -*• oo fiXtx)

_ f(Xtx)

f(tx) f(ktx) fifx)

f(fe)

ψ{λχ)

j

f(te)

f(tx)

0 0 , there exists ίο > 0 such that \L(tx,)/L(te)

— 1| < ε

(1.1.9)

for t > to- Let k be chosen in such a way that tk > to, then, upon setting t = tk in (1.1.9), we obtain \L{cktk)/L{tke)

-

\ \ < ε,

which contradicts (1.1.8). Let / € Ä 2 ( r ) a n d ^ = H e ( f ) . We set L(x) We observe that L{ta,)

_ 0 such that J2j=i Jh = ^ C ( / ' i , . . . , ik) are some constants. To complete the proof it suffices to say that (1.1.11) immediately follows from (1.1.15) and (1.1.16). • LEMMA 1.1.2. Let a cone Γ be closed, then for any function L € 72(Γ) there exists a real a > 0 such that 0 < inf L(x) x e A

< sup L(x)

< oo

(1.1.17)

xeA

for an arbitrary bounded set A c {χ; .γ e S, |x| > a). PROOF. By virtue of Theorem 1.1.2, for an arbitrary 0 < ε < 1 there exists b > 0 such that L ( | x | e ) ( l - ε) < L(x) < L ( | x | e ) ( l + ε) for |x| > b. Since the function g(t) = L(te) of one variable is slowly varying at infinity and Lemma 1.1.2 is known to be true for η = 1 (Seneta, 1976), from the last inequalities we find that (1.1.17) holds for some a > b. • Let Z " denote the set of all vectors in R" with non-negative integer coordinates. For k = (k\,..., k„) 6 Z " , k φ 0, and an infinitely differentiable function / ( x ) in R" we set

6

1. Tauberian theorems

where j — k\ + · · · + k„ and kjl,..., im < n, a n d f(0Hx)

1 oo. The statement concerning L** is proved in the same way as above. • The theorem below is known in the one-dimensional case as the integral representation theorem for slowly varying functions (Seneta, 1976; Bingham et al., 1987). THEOREM 1.1.4. The inclusion L e Τ2(Γ) holds if and only if there exist constants a > 0, c € R, and measurable functions η(χ), ε(ί) defined for χ € S, |x| > a and t > a, respectively, such that

L(x)

for χ e S,

> α, η(χ)

= exp ( η(χ)

+

c as |x| - > oo, and eik\t)

for any integer k > 0 as t —• oo.

=

m o(t~k)

9

1.2. Weak convergence of measures and functions

PROOF. Let L e Γ 2 ( Γ ) . We represent L as L ( x ) = exp(;/(x) + ln/(|x|)), where

assertions (a)-{d) of Theorem 1.1.3 are true for L2, assertion (c) of Theorem 1.1.3, η(χ) o(t~k)

0 as

for any k e Ν as t -*• oo, where v(t) (

= exp Ι η(χ) \

L(x)

and /(|x|) =

= In f i t ) . We set ε(ί)

+ v(a)

rM

+

s(u)

I

Ja

= tv'(t)\

—

then

du

U

for |x| > a, where the constant a is defined in Theorem 1.1.3, and integer k > 0 as t

By virtue of

L2(x).

-*• oo. By virtue of (1.1.15), v^k\t)

— o(t~k)

for any

oo. Now let us prove the sufficiency: for x, e e S

L

I

— —

= exp

Lite)

y

„ Λη(ίχ)

^ fM

I, η(ίβ)

+

/ Jtk

ι

u

J \ , du J

which yields

L{te)

\yH

^

J

as ί —^ oo. The theorem is thus proved.

1.2.

•

Weak convergence of measures and functions

In this section we assume that Γ is a closed convex acute solid cone with apex at zero. We r set G — int Γ. We introduce an order relation on Γ (Vladimirov, 1978): we write χ < y and r χ < y, if χ, y, y — x e Γ and xeT,y,y — xeG respectively. We omit the specification of the cone and write χ < y, χ < y where this does not lead to ambiguity. A real-valued function / ( x ) defined on an open set σ C Γ is said to be a monotone non-decreasing increasing., respectively) in Γ if f i x ) < fiy) x, yea.

The notation lim^j^ fiy)

there exists δ > 0 such that | fiy)

( f ( x ) > f(y),

— b (lim^f* fiy)

(non-

respectively) for χ
0

— b\ < ε for any y € σ,

- x\ < 8, χ < y iy
0 such that y + ζ - h e G for all h € R", \h\ < δι. Since Β is open, there exists 82 < 81 such that χ < u for all u € R", \u — z\ < 82· Then χ < u and y — u = y — ζ + ζ — u € G for all u e R", \u — z\ < 82', hence it follows that u e A for \u — z\ < 82. The lemma is proved. • LEMMA 1 . 2 . 2 . Let f

e Μι(σ)(/

G

Μ2(σ)).

(a) Then for any χ e σ there exist two limits Urn / 0 0 = / ( * - ) , y\x Urn f ( y ) = f(x+), y\x

(1.2.1)

= s u p ( / 0 0 , y e σ, y < x),

(1.2.3)

(1.2.2)

where fix-)

/ ( * + ) = inf(/(.v), y € σ, χ < y) ( f ( x - ) = i n f ( f { y ) , y < x,y e σ), f(x+) and Ax-) ( f ( x - ) > f i x ) > f(x+),

(1.2.4)

= s u p ( f ( y ) , x < y, y € σ), respectively),

< f i x ) < f(x+)

(1.2.5)

respectively).

(b) for f to be continuous at χ e σ, it is necessary and sufficient that fix+)

=

fix-)·

PROOF. Let / be monotone non-decreasing in Γ. Then by (1.2.3) and (1.2.4) for any ε > 0 there exist χε, ys € σ, χε < χ < yE such that ε > fix.)

- fixε)

Ε > f(ye)~ fi

x

> 0,

(1.2.6)

+ ) > 0.

(1.2.7)

By virtue of Lemma 1.2.1, the set Ue — {y\χε < y < ye} is open and non-empty, and therefore, so is the set Ve = Ue Π σ, because σ is open and x € Ve. Therefore, f(xe) < f i y ) < fix-) for any ye Ve,y < x, from the monotonicity of / and (1.2.3). Taking into account (1.2.6), we obtain l/OO - fix-)\

< e,

(1.2.8)

For y e Ve and y > x, it follows that f (x+) < / ( y ) < /iye) and (1.2.4). Taking into account (1.2.7), we obtain \fiy)-fix+)\x.

(1.2.9)

Since ε is chosen arbitrarily, (1.2.1) and (1.2.2) immediately follow from (1.2.8) and (1.2.9) respectively. We observe that f(z) < f i x ) < f i y ) for r, y e σ, ζ < χ < y. So (1.2.5) follows from (1.2.3), (1.2.4), and the last relation.

1.2. Weak convergence of measures and functions

11

Now let, in addition, f(x+) — fix-)· Then, because / is monotone, f(xe) < f ( y ) — f(ye) for y e VB. Taking into account relations (1.2.6) and (1.2.7), we find that | / ( x ) — fiy)\ < ε for y e Ve, because from inequalities (1.2.5) and fix-) = f(x+) it immediately follows that f(x~) = f ( x ) = f(x+)· Conversely, let f i y ) be continuous at a point χ e σ. Then from relations (1.2.1) and (1.2.2) just proved we find that fix-) = f i x ) = f(x+). If / is monotone non-increasing, then —/ is monotone non-decreasing, and all statements of Lemma 1.2.2 concerning / immediately follow from the fact that they are true for —/. The lemma is thus proved. • For i — 1,2 we set Μ,+(σ) = { / : / € Μ,·(σ), / ( * + ) = f i x ) Vx e σ}, Μ+(σ)

= Μι(σ)ϋΜ2(σ).

We say that a sequence fk e Λ/ ; + (σ) weakly converges as k -*• oo to / e Μ ; + ( σ ) and write fk => f

k ->· oo,

if fk(x) —> f ( x ) as k —• oo at an arbitrary continuity point x of the function / . LEMMA 1.2.3. If fk ^ f and fk g as k —*• oo, then f ( x ) — g(x)for χ 6 σ. PROOF. We consider the discontinuity points of the function f in the set lx = {y: y 6 a, y — tx,t > 0} for some χ e σ. There are at most countably many of them, because, as we easily see, the function • μ , k -*• oo, if μ/dA) μ{Α), as k -*• oo, for any A e 21 such that μ(3Α) = 0 (here and below dA stands for the boundary of the set A). We also say that the family of measures Μ is weakly relatively compact or weakly pre-compact if it is possible to extract a converging sequence from any sequence of measures in Μ . As concerns the weak convergence of measures, statements similar to those proved above for monotone functions are true. Here we give only one (see (Feller, 1966, Chapter VIII.6)). THEOREM 1.2.2. A family of measures Μ is weakly pre-compact if and only if sup μ(Α) < oo μ€Μ for any A e 21. The expression 'measure U on Γ" will mean that the measure U is concentrated on Γ. For a set A c R" and ί > Owe set tA = {y: y = tx, χ € Λ}. Let a measure U be given. We introduce the family of measures (Φ,, t > 0): Φ,Μ =

(1.2.21) pit)

where p(t) is a regularly varying function of one variable, ind ρ — γ > 0, A e Λ. We set A, = {x:x e R", |jc| < ί},

Β, = { χ : X e R", \x\ < t},

t > 0.

LEMMA 1.2.4. For some measure Φ, let Φί => Φ,

t ^ oo.

(1.2.22)

Then Φ(&4„) =

= 0

for any u > 0. PROOF. For any u > 0

1ΐηιϊηίΦ,(,4„) < Φ i A u ) < Φ(£«) < ümsupΦ,(Ä I < ). '-•οο t—•oo

(1.2.23)

Since the boundaries of the sets At and Au do not overlap for u ψ t, there always exists ν > 0 such that Φ(9Αυ) = 0, and hence, Φ(3Βυ) = 0. Then by the definition of weak convergence of measures lim Φ , ^ « ) = Φ(Λ υ ) - Φ(Β υ ) = Um Φ , ( 5 υ ) .

t—»oo

t-*oo

(1.2.24)

1. Tauberian theorems

16

By virtue of (1.2.21), for any u > 0 /u

p(t^)

N

Ι ΐ π ι ΐ η ί Φ , ^ ) = liminfΦ; ( - Λ υ ) = Uminf ^ - ^ Φ , «υ 04 υ ) /-»•οο

ι-* oo

\V

'

t-+ oo

P(t)

= ( - V lim Φ,04 υ ), \V/ t-+ oo limsup Φ^Ι?,,) =

(1.2.25)

üm Φ,(5„).

t—*oo

t-*oo

From relations (1.2.23H 1.2.26) it follows that lemma is proved.

(1.2.26)

Φ(Αα) = Φ(Bu), that is, Φ(dAu) = 0. •

Instead of (1.2.21), (1.2.22) we will sometimes briefly write

U(t·)/p(t) Φ, t oo.

1.3. Multidimensional Tauberian theorems of Karamata type Let Γ be a closed convex acute solid cone in R" with apex at zero (see the opening of Section 1.1). We set r = r*s{j:j€R",(>',x)>0

Τ dual 0. Laplace transform

Vx € Γ}.

The cone is said to be the cone to the cone Γ. The cone Γ is a closed acute convex solid one as well (see (Vladimirov, 1978; Vladimirov et al., 1988)). Therefore, C s int Τ φ The and the measure U on Γ are labelled, respectively, μ (λ) and U (λ):

Laplace-Stieltjes

= j e~(k'x)u(x)dx, £/(λ) = J e~(k'x)U(d τ provided that they exist for λ > a with some a € Τ (we recall that the last inequality means "(λ)

that λ — a e C, see the beginning of Section 1.2). For the sake of convenience, all measures are assumed to be concentrated in Γ. We also set G = int Γ, and let A denote the closure of a set A c R". The notation i « Γ means that A c G = int Γ. Let |/1| stand for the Lebesgue measure of a set A c R". For two functions f(x) and g(x) defined for .x e Γ, |x| > a, for some a > 0, the notation f(x) ~ g(x) as |x| oo means that

m

, ,,

g(x) Let a measure U and a function u(x) > 0 locally integrable in R" with respect to the measure be given; then denotes the measure defined by the equali

U

u(x)U(dx) V V(A) = f u(x)U(dx)

for every bounded measurable set / I C R " . The theorem below extends the well-known uniqueness theorem for one-dimensional Laplace transforms of measures (Feller, 1966, Chapter XIII, Section 1, Theorem la).

1.3. Multidimensional Tauberian theorems ofKaramata

type

17

T h e o r e m 1 . 3 . 1 . For measures U and V, let there exist their Laplace ~ τ and V(k)for λ > a with some a e T, and

U(k) = V(k) Then U =

VX >

transforms

U(k)

a.

V.

Theorem 1.3.1 follows from the inversion formula for Laplace transforms (Vladimirov et al., 1988, Chapter I, Section 2.5, formula (5.3)). We give an independent proof. Proof.

Let Η = Ü(z)

=

R"

+ i(a

J

e

i ( z

'

+ C) x )

=

{:: ζ = χ + i y , χ

U(dx),

V(z)

=

e R",

y > a},

ei{z-x)V(dx),

J

ζ €

Η.

The functions U(z) and V{z) are analytic functions in the domain Η (see (Vladimirov, 1964)). By the hypotheses of the theorem, Ü(k) = Ρ(λ) = ω(λ)

VA > a.

With the use of the uniqueness theorem for analytic functions of many complex variables (Shabat, 1992, p. 32), we see that Ü(z) = V(z). Therefore, J

e

( i v

'

x )

e~^'

x )

U{dx) =

e

J

( i v

'

x )

e~

( X

'

x )

V { d x ) Vv e R " ,

λ > α,

hence it follows that

J where

e

( i v

'

x )

=

Ui(dx)

J

e

( i v

'

x )

Vt(dx),

=

—— ω(λ)

„-(λ,ΛΓ) Ui (dx)

=

—— ω{λ)

(1.3.1)

„—(λ,χ) U(dx),

V\ (dx)

V(dx).

But U\ and V\ are probabilistic measures, and (1.3.1) implies that their characteristic functions coincide, hence U\ = V\. Thus, J

e-&*)u(dx)

=

J

e~

( k

'

x )

V(dx)

(1.3.2)

for any bounded Borel set A. Furthermore, the measure U is absolutely continuous with respect to the measure V. It is easy to see, indeed, that if Β e A is chosen in such a way that V(B) = 0, then by (1.3.2) J

e~

( X

'

x )

U(dx) =

0,

1. Tauberian theorems

18

and therefore, U(B) = 0, because e > 0 for any χ e B. Consequently, we obtain U(dx) = f(x)V(dx) for some function f(x). Then from (1.3.2) it follows that J

e~(X'x) f(x)V(dx)

= J

e~a'x)V(dx),

or f (x) = F-almost everywhere due to uniqueness up to a set of F-measure zero of the Radon-Nykodym derivative of the measure V(dx). Hence it follows that f(x) = 1 F-almost everywhere, and therefore, U = V. The theorem is proved. • By the restriction of a measure U defined on 21 to a Borel set σ C R" is meant the measure V defined by the equality V(D) = U(D) on the (5-ring of all sets D € 51 such that DQa. The theorem below extends the well-known continuity theorem for Laplace transforms (Feller, 1966, Chapter XIII, Section 1, Theorem 2a) to the multidimensional case. THEOREM 1.3.2. Let (Uk, k e N ) be a sequence of measures on Γ. ~ τ (1) If for some a & Τ there exist the Laplace transforms Uk("k) VA > a, k e N, and Ük(X)

ω(λ) < o o

VX > α,

(1.3.3)

as k —> oo, then ω{λ) is the Laplace transform of some measure U on Γ, and Uk => U as k -*• oo. (2) Conversely, if Uk ^ U as k oo and Uk(a) is bounded, then relation (1.3.3) is true, and for the measure U there exists the Laplace transform U(λ) = ω(λ) for T λι > a. PROOF. We begin with proving part 2. We take a sequence ijt t 0 0 such that U(dAk) = 0, where Ak = {χ: χ e R", |x| < tk}. Let uj^m) and U(m) be the restrictions of the measures Uk and U, respectively, to Am. By the definition of weak convergence of measures, U^ for any m € Ν as k oo, therefore (Bender, 1974), as k -*• oo f e~(X'x)Uk(dx) J Am Τ Therefore, for λ > a f e~(X'x)Uk(dx) JΓ

f e~(k'x)U(dx), * Am

= mUm [ ->°°JAm

Vm e Ν,

λ > a.

(1.3.4)

e'^Uidx)

= Um lim f

e~(X'x)Uk(dx)

< UmsupÜ k (a) < oo.

(1.3.5)

Τ For any k, m e Ν, λ > a \Ük(X) - Ü(λ)I < ί e~a'x)uk(dx) JBm +

+ f e~(X'x)U(dx) JBm

f e~{X'x)Uk(dx) IJAm

- f e~^'x)U(dx) JAm

,

(1.3.6)

1.3. Multidimensional Tauberian theorems ofKaramata type

19

where Bm = Γ \ Am. For h = λ — a, as we easily see, ί

< sup

e-(k'x)Uk(dx)

J Bm


τ|χ| inffe € fi(a, b) > 2 k ~ l c , where c — i n f b ) > 0, for χ e A k € N, because a e C = int Γ * . As shown in Lemma 1.2.4, Φ(3Ζ)) = 0, therefore, U(tD)/R(t) For t > t0, let U(tD) Ü(τα)

< U(tD)

< KR(t), +

Φ(ΰ),

t

oo.

where Φ(Ζ>) < Κ < oo, then for t > t0 e~c2k~l

U(2ktD)

< K(R(t)

+ ^

R(lkt)).

fceN fceN

Hence it follows that ΙΙ(τα)/R(t) are bounded (Feller, 1966, Section ΧΙΠ.5, relation (5.11)). If (1.3.9) is true, then (1.3.8) and (1.3.10) immediately follow from part 1 of Theorem 1.3.2. The proof of the theorem is thus complete. • THEOREM 1.3.4. For some function u(x) u(k) < oo for any λ e C. (1) IfR(t)

>

0 defined and measurable on S, let

is regularly varying at infinity, u(rk)/R(t)

ψ(λ)

< oo,

t —> oo,

λ e C,

(1.3.11)

and u(x) — f(x)g(x), where f e Ri(G) (see Section \A), and g is monotone in the domain G (see Section 1.2), then for χ e G u(tx)tn/R(t)

ψ(χ) < oo,

t-+oo,

(1.3.12)

and there exists a measure Φ on Γ such that Φ (dx) = -n, then (1.3.11) and (1.3.13) are true with φ = Ha(u) and R{t) = tnu(ta) for any a e S (see Section 1.1), and the relation Φ(9Γ) = 0 certainly holds. (3) Let R(t) be a regularly varying at infinity function, and let (1.3.11) hold. If (u{tx,) - u(tx))tn/R(t)

0

as t -*• oo for any vectors xt,x e G such that x, —• x, then (1.3.12) and (1.3.13) hold for some function φ on G and some measure Φ on Γ. REMARK 1.3.1. ΙίΦ(ΘΓ) = 0 in Theorem 1.3.4, then by virtue of (1.3.13) the equality ψ{λ) — φ(λ) < oo holds for any λ e C. PROOF. W e set U(dx)

= u(x)

dx.

Let the hypotheses of part 1 be satisfied. Then by virtue of Theorem 1.3.3 there exists a measure Φ on Γ such that U(t·) Φ, t oo. R(t) Thus, U(tA) _

ί

u(ty)t"

m ~ L m dy

Φ(^),

t

oo,

(1.3.14)

for an arbitrary set A e % such that Φ(&4) = 0. We take an arbitrary vector a e S and set , , , ht(y) =

g{ty)t"f(ta) ^ ,

y ε G,

t>

o.

For any t > 0 the function ht(y) g M+{G) because g G M+(G). Let us demonstrate that for any χ € G there exists a constant cx < oo such that for sufficiently large t h,(x) < c , .

(1.3.15)

If g does not decrease, then Φ φ ) = Um f

" ^ d y

> lim sup f i^\ht(y)dy t-*OO JDI fita)

> |£>,| inf ^ \ h t { x ) , yeD f(ta)

(1.3.16)

for an arbitrary ball D with centre at the point Oc such that D c G and Φ(3£>) = 0, where τ D\ — {y\y G D, y > x}. Let Ha(f) = ψγ. Then by virtue of Theorem 1.1.2, since the function ψ\ is continuous and positive, there exist b > 0 and to > 0 such that • f / t o ) -> u η inf > b > 0 yeD f(ta)

1. Tauberian theorems

22

for all t > to', hence it follows that (1.3.15) holds for t > to with cx = increase, then the same reasoning yields

· If g does not

Φ ( £ » ) > | Ζ ) 2 | inf ^ \ h t { x ) , yeD / ( ί α ) where D2 = {y : y £ D, y < x}, therefore, (1.3.15) holds for t > to with cx = • By virtue of Theorem 1.2.1, the set of functions {ht(x), t > ίο} is weakly relatively compact. As t^ ΐ 00, let h,k

k^-oo.

(1.3.17)

Let us demonstrate that the function ψ\(x)h(x) is the density of the measure Φ with respect to the Lebesgue measure in G. Let the set A Γ, A e SI. Then there exist ε > 0 and r r λ > 0 such that εα < χ < λα for any χ € A. It is easy to see, indeed, that if for arbitrary ε > 0 there exists χ e A such that χ — αε 6 Ε — R" \G, then there exist sequences x/c -> x e A and -> 0 as k 00 such that Xk — αε^ e E. By passing to the limit as k - > 00, we find that χ e Ε because Ε is closed, which is impossible. Similarly, if αλ — χ € Ε for any λ > 0 and some χ € A depending on λ, then there exist sequences λ^ - » 00 and Xk -*• χ € A such that αλk — Xk e Ε or a — e E. If k tends to infinity, we see that a € E, which is impossible. Thus, there exist vectors u,v e G such that u < χ < ν for any χ e A. By virtue of Theorem 1.1.2, there exist t\ > 0 and a constant c > 0 such that f(tx) < c f(ta)

for t > t\ and χ e A. In view of relation (1.3.15) and the monotonicity of ht, h,(x) max(c u , cv) for sufficiently large t > tz and χ 6 A. Therefore, u(tx)tn f(tx), , , ——— = ——ht(x) < cmax(c„,c„) R(t) f(ta)


max(?i, ti) and χ e A By virtue of the Lebesgue theorem, from (1.3.17) and (1.3.18) we obtain f u(tk)tI / ' f dx JA R(tk)

f / 0). From the last bounds it follows that U(tA)

f

Ryt)

JA

/

„

l i m s u p — — — < I ε.

Without loss of generality wc assume that xm, ym —> χ € Κ. as ftt —> oo, tti £ L. Then Ihm(ym)

- hm(xm)I

< Ihm(ym) - hm(x)\

+ \hm(xm)

as m -*• oo, m e L, because the sequence (hm(x)) contradiction proves the lemma.

- hm(x)\

0

is asymptotically continuous. This •

THEOREM 1.4.1. Let a sequence (hm(x)) be asymptotically continuous in D. In order for it to be pre-compact in D in the pointwise convergence topology, it is necessary and sufficient that for any χ e D limsup |A m (x)| < oo. m-* oo

(1-4.2)

PROOF. It is clear that condition (1.4.2) is necessary. Let us prove its sufficiency. Let (1.4.2) hold and D' — { χ ι , χι, X 3 , . . . } be a countable everywhere dense set. With the use of Cantor's technique we construct an unbounded subset L c Ν such that for all χ e D' hm{x)

-*• h(x)>

m

00,

m e L,

(1.4.3)

for some function h(x) on D', where \h(x)\ < 00. It is easily seen that h(x) is uniformly continuous on the set Μ — Κ Π D' for any compact Κ C D. By virtue of Lemma 1.4.1, for any ε > 0 there exist / e Ν and S > 0 such that Ihm(x) - hm(y)\

< ε

(1-4-4)

for any x, y e K, r(x, y) < 8, m e N, m > I. Now let x, y e Μ. If m grows without bound in (1.4.4), recalling (1.4.3) we see that \h(x)-h(y)\ 0 in such a way that the set K(x, 3) = {j: y € D, r(x, y) < 5} is a compact in D. Then from the uniform continuity of the set h on the set K(x, I such that |Μβ)-Α(β)Ι k, m € L. Then I M * ) - Kx)\

< I M * ) - M f l ) | + IΛ«(β) - h{0)I + Ih(a) - A(*)| < ε

for any m e L,m > k, which is the desired result. The theorem is proved. THEOREM 1.4.2. We assume that a sequence (hm(x), tinuous in D and that M*)

-»· h(x),

me

•

N) is asymptotically con-

|A(*)| < oo,

(1.4.6)

for all χ € D as m oo. Then the function h(x) is continuous in D and relation (1.4.6) holds uniformly in χ e Κ for any compact Κ c D. PROOF. The continuity of h(x) follows from relations (1.4.1) and (1.4.6). We assume that there exist a compact Κ c D, ε > 0, a sequence xm € K, and an unbounded set L c Ν such that \hm(Xm)-h(xm)\>B

(1.4.7)

for any m e L. Without loss of generality we take xm —χ € Κ. Since Ihm{xm) - h(xm)I < Ihm{xm)

- hm(x)\ + \hm(x) - h(x)\ + \h(x) -

h(xm)|,

in view of the asymptotic continuity of hm, relation (1.4.6) and the continuity of h{x) we obtain Ihm(xm) — h{xm)I

0,

which contradicts (1.4.7). The theorem is proved.

m

oo,

m e L, •

1.4. Weakly oscillating ftinctions

THEOREM 1.4.3.

Let D be connected

limsup sup

and (hm(x)) \hm(x)

29

be asymptotically

- hm{y)\

continuous.

Then

(1.4.8)

< oo

m—Hx> x,y€K for any compact

Κ c.

D.

P r o o f . We take arbitrary a,b e D. Let {x(t), t e [0,1]} be a continuous curve which joins the points a and b, x(0) = αΛχ(1) = b. We set Τ = {t:t

€ [0,1], l i m s u p \ h m { a )

— hm(x(t))\

=

oo}.

m—foa

Let us show that Τ = 0 . Let the contrary be true, that is, Τ φ 05, and let ν — inf T. It is clear that ν > 0. In view of continuity of the curve x(t) and asymptotic continuity of (hm), we obtain hm(x(s))

— hm(x(t))

—> 0 ,

m

oo,

s , t -*• v.

Therefore, there are k 6 Ν and oo

\hm(a)

—

(1-4.9)

We take arbitrary t € [υ — S, v) and

/! m (x(s))| < 1 + limsup m—>oo

\hm(a)

- hm{x{t))\

< oo,

which contradicts our choice of s, s e Τ. Thus, Τ = 0 . Hence limsup|/z m (a) m—>oo

hm(b)\


·οο,

m ->· o o ,

m €

L.

Without loss of generality we assume that xm —> a G Κ, ym —y b € Κ as nt —> oo, nt β L. By virtue of the asymptotic continuity of (h m ), hence it follows that \hm(a)

— hm(b)\

oo,

m o o ,

m e L,

which contradicts (1.4.10). The theorem is proved. D e f i n i t i o n 1.4.2. A family of functions (ht(x), continuous

in D a s /

o o if f o r a n y χ e ht(y)

—h

t

• t

> ίο) is said to be

asymptotically

D

( x ) 0 ,

t ο ο ,

y

χ.

R e m a r k 1.4.1. It is clear that, in view of Definition 1.4.2, Lemma 1.4.1, and Theorems 1.4.1,1.4.2,1.4.3 remain true after changing m 6 Ν for 'continuous' parameter t > to-

30

1. Tauberian theorems

As above, let Γ be a cone in R" with apex at zero, S = Γ \ {0}, and let t be a nonnegative variable. DEFINITION 1.4.3. A non-negative function / ( x ) defined for χ e Γ, |x| > a > 0, is called feebly oscillating (at infinity in Γ) if there exists a vector e € S such that f(te) > 0 for all sufficiently large t, and for arbitrary χ e S, as t -*• oo,

f ( t x t ) - f ( t x ) = o(f(te)), provided that x, e S, xt

(1.4.11)

x.

Unless otherwise stated, in what follows we assume that S is connected. For illustrative purpose, we give a series of examples of feebly oscillating functions for η = 1, (here Γ = R+ = {t,t > 0}). (1) Let a function f i x ) > 0 be defined and differentiable for χ > a > 0, and for some real α, β let a f ( x ) < χf i x ) < βf{x). Then / is feebly oscillating at infinity (such functions were introduced in (Keldysh, 1973) and then used in Tauberian theorems, see, e.g., (Selander, 1963)). (2) For all c > 1, let there exist constants α, β, and ν > 0 such that c-'(x/y)a < m / f i y )
y > v. Then / is feebly oscillating at infinity. For monotone functions, one of the last inequalities is certainly true (such conditions on monotone functions / were used in Tauberian theorems proved in (Belogrud, 1974; Matsaev, Palant, 1977; Sultanaev, 1974); see also the monograph (Kostyuchenko, Sargsyan, 1979)). (3) Let a function f ( x ) > 0 do not decrease, and for all λ > 1 let f(kt) lim sup - τ - — = φ(λ) < oo, t—•oo f i t ) while 0,

ρ(λ) < oo,

for any λ > 0 as t oo. Then f is feebly oscillating at infinity. Such functions were first introduced by J. Karamata (Karamata, 1930a; Karamata, 1930b; Karamata, 193 la) in connection with establishing extensions of a known Tauberian theorem due to Hardy and Littlewood (Hardy, Littlewood, 1914). Now regularly varying functions enjoy wide application, see (Bingham et al., 1987).

1.4. Weakly oscillating functions

31

We observe that feebly oscillating functions may oscillate between two power functions. For example, the function xsiDlalnx

f{x) =

oscillates between x~x and λ: but very slowly, so remains in the class of feebly oscillating functions. In passing it should be mentioned that the regularly varying functions oscillate between χρ~ε and χρ+ε for any fixed ε > 0, where ρ is the index of regularly varying function (Seneta, 1976; Bingham et al., 1987). The functions in the above examples 1-3 are, generally speaking, not regularly varying, as seen from the function f(

X

) =

x 2+smlnln*

For feebly oscillating functions, the following theorem is true. THEOREM 1 . 4 . 4 . Let a function compact

f be feebly

lim sup sup i-coo xeK and the

oscillating

at infinity

in Γ. Then for

any

Κ c. S

A

t e

< oo,

(1.4.12)

)

limit y

lim

If i t y ) - f i t x )1

sup

——

x,yeK 00 1 x-y\ 0 for t > t0. For χ e S and t > i 0 , w e set fitx) ht(x)

= Ate)

(without loss of generality, we let / be defined in the whole S). Let us demonstrate that the family of functions (h t (x), t > to) is asymptotically continuous in S as t -h>- oo (see Definition 1.4.2). We assume the contrary: let there exist sequences tm oo, xm € S, xm —>xeS as m—too and ε > 0 such that for all m e Ν \ f ( t

m

x

m

) - A t

m

x ) \

> e

Atme)

We also assume that ίο < h < h < thatx(i) -»· λ: as t oo. By (1.4.11),

We set x(t) = xm for t e [tm,tm+1). It is clear

f{tx{t)) - f{tx) = oifite)),

t

oo.

m

> oo,

Setting t — tm in the last equation, we obtain fitmXm)

- fit

m

x)

- o(f(tme)),

32

1. Tauberian theorems

which contradicts (1.4.14). Thus, the family of functions ( h t { x ) , t > to) is asymptotically continuous in S . Therefore, from Theorem 1.4.3 (see Remark 1.4.1) it follows that for any compact Κ c S lim

sup

< oo.

fite)

(1.4.15)

Without loss of generality, we assume that e e Κ (the compact K, if needed, may be extended by one point). It is clear that f(tx) \f(tx)-f(te)\ jy_> < uk , j κ >\ + j (1A16) fite) fite) Now (1.4.12) follows from (1.4.15), (1.4.16) because e € K. Relation (1.4.13) follows from Lemma 1.4.1 and asymptotic continuity of the family of functions ( h t i x ) , t > to)· The theorem is proved. • REMARK 1 . 4 . 2 . F r o m ( 1 . 4 . 1 1 ) it f o l l o w s t h a t f o r all λ > 0, xt,x

€ S, xt

χ as

t -> oo f{txt)

-

f(tx)

m e )

•

Ii is easily seen indeed that f{txt)

- f(tx)

=

/((ίλ)(χ,λ~1))

m e )

-

fdtXXxX-1))

f(tke)

as t ->• oo by (1.4.11). COROLLARY 1.4.1. If f is feebly oscillating ((1.4.11) holds), thenforallO

Q

)

for some c > 0 and all t large enough by (1.4.12). If S is not connected and / feebly oscillates in Γ, then (1.4.12) may be broken. For example, let Γ = R 1 , f ( x ) = 1, χ < — 1; f ( x ) = l n x , x > 1. It is easy to see that / obeys (1.4.11) with e = — 1, that is, / is feebly oscillating, but as t - * oo fit) . , = In? —> oo, fite) in other words, (1.4.12) does not hold. Let us give a definition of weakly oscillating functions.

1.4. Weakly oscillating functions

33

DEFINITION 1.4.4. A function / ( x ) , which is defined and positive for χ e Γ , |x| > a > 0, is said to be weakly oscillating (at infinity in Γ ) if

-* 1

f(txt)/f{tx) for all xt, χ 6 S, xt ->· x, as t

(1.4.17)

oo.

REMARK 1.4.3. In the definition of a weakly oscillating function it suffices to consider (1.4.17) to be true for all χ 6 S, |x| = 1. It is easily seen indeed that if |x| =

1 and (1.4.17) holds, then for any positive ν

f(tvx,)/f(tvx)

1,

= f((tv)x,)/f((tv)x)

t - * oo.

THEOREM 1.4.5. Let a function f be weakly oscillating at infinity. Then (1) the function f is feebly oscillating and (1.4.11) holds for any vector e e S. (2) the relations 0 < liminf

f{tx) f(tx) — — - < lim sup sup — — - < oo

inf

oo

f(ty)

x,yeK

(1.4.18)

f(ty)

hold for any compact Κ c S; the limit lim

f(ty)

sup x >y* K

t—>00 L I x-y\ 0 and sequences t^ f oo, Xfc, jfc, X/t —> x, —>• χ as A: oo, such that I f{tkyk)/f{tkXk)

- 1| > ε > 0.

If we now set a(t) = xk and b(t) — yk for t € \tk, ifc+i), then we see that for t = tk \f(tb(t))/f(ta(t))-\\>e>Q. But in view of (1.4.17) f{ta(t))

f(ta{t)) /(fx)

f{tb(t))

f{tx) f(tb(t))

1,

t

oo.

This contradiction proves (1.4.19). W e take arbitrary vectors x , y e S. According to (1.4.19), there are ε > 0 and to > 0 such that

l/(ta)//(tb)-ll to, where I =

{:::

— x{t),

t e [0,1]} is a continuous

curve which joins the points χ and y while x ( 0 ) = x , x ( l ) = y, I c S. W e choose me

so that w{\/m) < ε, and set xi = x ( l / m ) ,

X2 = x(2/m),

...,

xm-\

— x((m -

l)/m),

Ν

1. Tauberian theorems

34

where w(8) is the modulus of continuity of the curve x(t). Then m

" f{tXl)

•' -

' - ·

(1.4.20)

f{ty)

We assume that (1.4.18) does not hold. Then there exist sequences tk such that as k —> oo f(tkXk)/f(tkyic) oo.

oo, Xk, yk

e

Without loss of generality we assume that Xk —*• x € K, yk -*• y € Κ as k —oo. virtue of (1.4.19), /fa**) ? χ /fay*) ) 1 /fa*) ' f(tky) as k —*• oo. Therefore,

/fa*)

/fa**) / f a * ) /fa}*)

/fay)

/fay*) /fa**) /fay)

K,

By

oo

as k -»· oo, which contradicts (1.4.20). Thus, (1.4.18) is proved. Let xt,x,e € 5, x r -»• Λ: as / oo. By (1.4.18), there exist c > 0 and ίο > 0 such that for t > i 0 f ( t x ) / f ( t e ) > c. Then for t > to \ f ( f x t ) ~ /(?*)! < |/(*st) ~ /(?*)! /(fe) ~ c/(f*)

1 /(?*,) _ j

=

c

From the last relation it follows that ( 1 . 4 . 1 1 ) holds true for an arbitrary vector e € S. The theorem is proved. • With the use of Theorem COROLLARY 1 . 4 . 2 .

1.4.5,

we arrive at the following.

The assertions below are equivalent.

(a) A fiinction f is weakly oscillating. (b) (1.4.11) holds true for any vector e e S. (c) (1.4.11) holds true for any vector e e S, \e\ = 1. (d)

(1.4.11)

holds true for some vector e € S, and l i m i n f ^ O t->oo f(te)

for any χ e S, |λ:| = 1. We recall that a cone Γ is said to be solid if int Γ φ 0 . C O R O L L A R Y 1 . 4 . 3 . Let f be feebly oscillating at infinity in Γ ( ( 1 . 4 . 1 1 ) holds). Then there exists a solid cone R with apex at zero such that e e int R and f is weakly oscillating at infinity in Γ Π R.

35

1.4. Weakly oscillating functions

PROOF. Let / obey (1.4.11). Then by (1.4.13) there exist δ > 0 and t0 > 0 such that < 1/2

\f(tx)/f(te)-l\

(1.4.21)

for all t > i 0 , χ e S, \x - e\ < 8. We set R

=

{y:

y =

λχ,

χ

0},

where e R", \x-e\

Β = {x:x

< 5}.

It is clear that R is a solid cone, e e int R. We choose an arbitrary vector y e S Π R, y = λχ for some λ > 0 and χ € Β. By virtue of Corollary 1.4.1, there exists t\ > 0 such that > c > 0

f{tXe)/f(te)

(1.4.22)

for all t > t\ with some constant c. From (1.4.21) and (1.4.22) it follows that f(ty)

=

f(te)

fjtkx)

fjtXe)

^ c

Atke)

f(te)

~ 2

for t > max(?o/λ, t\). In order to complete the proof of Corollary 1.4.3, it remains to make use of Corollary 1.4.2. • It is clear that in the one-dimensional case the feebly oscillating functions are weakly oscillating. The example below demonstrates that this is not the case for η > 1. Let Γ = {χ = (χι,Χι), Χι > 0, χι > 0},

where* = (x\,x2)

U

*>>0,

I mx2,

x\ = 0,

6 Γ. It is not difficult to see that / obeys (1.4.11) withe = (1, l),that

is, / feebly oscillates. But f o r * = (0,1) f(tx)/f(te)

= 1ηί/ί -*• 0,

t

oo,

and therefore, / is not weakly oscillating. The theorem below extends the well-known integral representation theorem on regularly varying functions in (Seneta, 1976). THEOREM 1.4.6. If a cone Γ is closed, then (1) a function f is feebly oscillating at infinity in Γ if and only if it admits the representation f ( x ) = h(x) exp ^ j f ' ' ^ y - d t ^ , where χ e Γ, |x| > b > 0, and h(x)

is feebly oscillating at infinity in Γ, s(t) is

measurable, and sup h(x) \x\>b

< oo,

(1.4.23)

sup|e(?)| < oo. t>b

36

1. Tauberian theorems (2) A function fix) resentation

is weakly oscillating at infinity in Γ if and only if it admits the rep-

PH

f ( x ) = exp ηix) + /

dt

,

(1.4.24)

where |x| > b > Ο, χ € Γ, and sit) is measurable, sup \x\>b

< oo,

sup |ε(ί)| < oo, t>b

and the function exp(//(x)) weakly oscillates at infinity in Γ. PROOF. For n = 1 the theorem follows from the corresponding assertion concerning the RO-varying functions (Seneta, 1976, Theorem Al). We recall that a positive and measurable for χ > a > 0 function / ( x ) of one variable is said to be RO- varying at infinity if for any λ > 0 0 < hminf

f(x)

< hmsup

/(χ)

< oo.

Let n > 1 and the function / be weakly oscillating at infinity in Γ. We choose some vector e e S and represent / as f i x ) = /(\x\e) 77nV(Wi;)· In view of closeness of Γ, the set Β = {χ: χ e Γ, |x| = 1} is a compact. From this fact and Theorem 1.4.5 it follows that .. . f /(χ) ^ < r 0 < limmf . s < hmsup „„ . x < oo, |*Koo /(\x\e) |χΚοο fi\x\e) and the function / ( x ) / / ( | x | e ) is weakly oscillating as the ratio of two weakly oscillating functions. It remains, because, as we have seen, the theorem is true for n = 1, to represent fi\x\e) in the form (1.4.24). Part 2 of the theorem is proved; part 1 is validated similarly. n

•

COROLLARY 1.4.4. Let a cone Γ be closed and a function f (x) be weakly oscillating at infinity in Γ. Then there exist real a, ß, b > 0, c > 0, CQ > 0 such that coM" < co\x\ß
1,

(1.4.25)

< c\x\",

Vx e Γ-.b/t < \x\ < 1

(1.4.26)

for all t > b and e e Γ, \e\ = 1. PROOF. In accordance with representation (1.4.24) f o r i > b, |e| = 1, |x| > 1,

(

7ΰ7)=&χρ{η(ίχ)~η(ί£) < c exp I β J

+

1

c'w ι \ -du 1 = cexp(/31n|x|) = c\x\ß

1.4. Weakly oscillating functions

37

with some constants c and β. If b/t < |x| < 1, then for t > b and \e\ = 1 fitx) fite)


0 | Vi > b, Wx e S, bit < |x| < 1, f i t x ) / f { t e )


|

Vi

>

b,

Vx

|x|

e S,

>

1,

f { t x ) / f ( t e )
a, such that for any compact Κ Q D fm(x)

-»· f(x),

m^-oo,

c uniformly in χ e K, and ω ( λ ) = / ( λ ) for all λ > a. c PROOF. We take arbitrary χ € D and λ > a. Then for sufficiently large m > k l/m(x)| < fm(x)-\A\~l

< ΜΓ1 J e-^\fm(x) < sup I fm(x) y€A

+ \A\~l I

jj-{X'y)fm{y)dy

- fm{y)\dy +

jj-{X'y)fm{.y)dy

ΙΛΓ'ΐΛ,Κλ)

+ I Α \ - ι φ ( λ ) < c < oo,

- fm(y)I

where A = {y\y e D, — x\ < ε}, \A\ is the Lebesgue measure of A, and ε > 0 is appropriately chosen. The sequence {fm(x), m e N } is thus pre-compact in D in the pointwise convergence topology (Theorem 1.4.1). For some unbounded set L c Ν and all χ e D let, as m —> oo, m e L, /(*)·

fm(x)

(1-5.4)

Then by virtue of Theorem 1.5.1 for m € L, as m -*• oo,

fmw

^

m

c for all λ > a. Therefore, / ( λ ) = ω(λ). Since / is continuous (Theorem 1.4.2), it is uniquely determined by its Laplace transform ω ( λ ) . Since the limit function in (1.5.4) does not depend on the choice of the subsequence L c N, (1.5.4) holds as m —* oo, m € N. From Theorem 1.4.2 it also follows that (1.5.4) holds uniformly in χ e Κ for any compact Κ c D. • First we prove the following Tauberian theorem of Littlewood type (Littlewood, 1910). THEOREM 1.5.3. Let a function r(t) be regularly varying at infinity with index γ > —η (see (Seneta, 1976)), a function f(x)

be r-slowly varying at infinity in Γ (see Defini-

tion 1.4.5), and for all λ € C let |/(λ)| < oo. Then the following assertions are true. (1) If ^ ^ tnr(t)

ψ(λ),

|*(λ)| < oo

(1.5.5)

-η. φ(χ),

\φ(χ)\ < oo

(1.5.6)

for all λ € C as t —*• oo, then f(tx)/r(t)

for all χ 6 S as t —> oo; further, for any λ € C there exists φ (λ), and φ(λ) = ψ(λ).

(1.5.7)

1.5. A multidimensional Tauberian comparison theorem

41

(2) If (1.5.6) holds, then (1.5.5) and (1.5.7) also hold with some function ψ (λ). (3) Under the hypotheses of assertions 1 and 2, the function 0 and χ e S), and relation (1.5.6) holds uniformly in χ € Κ for any compact Κ c S.

PROOF. We fix some ε > 0 so that γ — ε > —η. From the integral representation theorem for regularly varying functions it follows (Vladimirov, Zavyalov, 1981) that there exists ίο > 0 such that h(u) < r(ui)/r(t)

< g(u)

(1.5.8)

for all t > to and u > to/t, where uy+S

(u)=i

>

«>1+6. e

\ur- ,

u


l-ε,

«>l+e. u < 1 - ε,

and the functions g(u), h(u) take some constant values on the interval [1 — ε, 1 + ε]. First we prove assertion 2. By virtue of relations (1.4.28), (1.5.6), and (1.5.8), there exist t\ > to, c < oo such that |/Qx)| r(t)

for all ί >t\,x we set

| / ( f x ) | r(f|x|) ^ | / ( τ χ / | χ | ) r(i|x|) r(t) ~ r(r) t\ w _L 1/(^jc/[x|) — / ( r e ) j < —T-z-g( X ) + 7-T r(z) r(τ)

=

e Γ, |x| >t\/t,

* )
t\ and χ e Γ,

8 t i X ) =

{θ,

\x\
t\} is asymptotically continuous in S as t ->• oo (see Definition 1.4.2 and Remark 1.4.1), and in view of (1.5.6) and (1.5.9), for all χ e 5 as t —> oo gtix)

φ(χ),

and |g/(x)| < cg(\x\) for t > t\. There exists, obviously, the Laplace transform of the function g(|x|) in C. Hence, with the use of Theorem 1.5.1, for all λ e C we obtain £,(λ)

ψ(λ),

t

oo.

(1.5.10)

But as t -> oo §tiX) =

f ß g l J\x\>t,/t l\x\>ti/t nt) r(t)

/(λ/ο 'MO

e

- M

d x

fiy) [f J\y\• oo. By (1.5.13), to each Μ > 0 we put in correspondence some τ = τ(Μ) in such a way that for t > τ, t e T2 the formula fite)

(1.5.15)

fit) is true. Then by virtue of (1.5.8) and (1.5.15)

J\x\>z/t

fit)

J\x\>Tit >M

f J\x\>T/t

Κ'Μ)

fit) a x)

e~ ' h(\x\)dx.

43

1.5. A multidimensional Tauberian comparison theorem

Therefore, liminf 9ft f t^oo,teT2 J\x\>x/t

f(t xle)

\

e-^x>dx r{t)

> Μ f e~^x) Jr

h(\x\) dx

We set U — r(0). The inequahty is just weakened: liminf m [ t^,teT2 J |*|>f4/,

f(t X e} (X x) \ } e- ' dx

> Μ [ e~(X'x)h(\x\) Jr

r(t)

dx.

Since the left-hand side of the last inequahty does not depend on Μ , we see that

J\x\>t4/t

r(t)

which contradicts (1.5.14). Hence, 1 fite)\ limsup ——— < oo. /—•oo r(t) r

Therefore, as for assertion 2, we conclude that (1.5.9) is valid. The rest follows from Theorem 1.5.2. Assertion 3, in view of the abovesaid, is an immediate corollary to Theorem 1.5.2. • For two cones Γι and Γ2 we write Γι -< Γ2 if the closure of the set {χ: χ e Γι, |x| = 1} is contained in int Γ2. The main result of this section is the following multidimensional Tauberian comparison theorem. THEOREM 1.5.4. For some non-negativefiinctions f i x ) and g(x) defined in Γ, let there exist their Laplace transforms f (λ) and g(A) for λ e C, let the function f (x) be weakly oscillating at infinity in Γ (see Definition 1.4.4), let g(x) = r(x)h(x), where r(x) is monotone inside G (see the beginning of Section 1.2), and let the function h(x) be weakly oscillating in G and ind_ / > — η (see Remark 1.4.4). If g(Xt)/f(Xt)^\,

t-*

0+,

(1.5.16)

for some solid cone Co -< C and all λ e Co, then g ( x ) / f ( x ) ^ 1,

|x|-»00,

xer0,

(1.5.17)

for any cone Γο -< Γ. This Tauberian theorem extends those given in (Vladimirov, 1978; Stadtmüller, Trautner, 1979; Stadtmüller, Trautner, 1981; Stadtmüller, 1983). PROOF. We assume the contrary: let there exist a sequence xm \Xm I 00 as m 00, and a constant c φ 1, 0 < c < 00, such that g(xm)/f(xm)

-» c,

m - > 00.

e

Γο such that

(1.5.18)

1. Tauberian theorems

44

We set xm - tmem, where tm — \xm\, em — xm/\xm\· Without loss of generality, we assume that em e e Β = {χ: χ e Γ, |x| = 1} as m -*• oo. We observe that e e G , because Γο -< Γ. Further, for λ e Co f(k/t

m

) =

[ e-«< JΓ

x

" ^ f ( x ) d x = t"mf{xm) f - φ ^ - e - ^ d x . J Γ J \*mem)

(1.5.19)

We observe that the sequence of functions am(x)

= f(t

x)/f(xm)

m

is asymptotically continuous in S (in the sense of Definition 1.4.1). It is easily seen indeed that, by virtue of Theorems 1.4.5 and 1.4.4 , , / \ f(tmx)~f(t am{x) - am(y) = — f(tmem)

y)

m

, ^ J { t m x ) -f{t = (1 + o ( l ) ) — f(tme)

m

y) m = o(l)

M „ m (1.5.20)

as m oo, y x, x, y e S. Next, by virtue of Corollary 1.4.4, there are α, β, I > 0, c ι > 0, — η < a < β < oo, such that for all t > I and m e N f(tx)/f{te

m

) < φ{χ),

(1.5.21)

where

\d\x\o,

I/1 l/t

m

,

\x\ < l / t

m

.

( x ) , m e N } is asymptotically continuous

< φ(χ),

(1.5.22)

and since a > —n, we see that φ(λ) < oo for all λ e C. Taking account for Theorems 1.4.1 and 1.4.2, without loss of generality we assume that bm(x)

a(x)

(1.5.23)

as m - » oo for all χ € S and some continuous function a ( x ) . In this case, by virtue of Theorem 1.5.1, taking into account relations(l.5.22)and(1.5.23), we find that for allA e C as m oo bm(k)

-»· β ( λ ) < oo.

(1.5.24)

Further, in view of (1.5.24), f Jr

f(tmem)

= bm(k)

+

f JM h(t m

r(tmx)h(tme) "«W

= ~—77—: f(xm)

·

It is clear that the functions um(x) are monotone inside G and the sequence of functions {vm(x), m e N} is asymptotically continuous in G. By virtue of (1.5.27) and Theorem 1.3.2, as m -» oo

Jf um(y)vm(y) dy^ J a(y) dy,

(1.5.28)

where r

A = {y\y e Γ, y > χ, \y - x\ < ε}. We observe that um(x) < c 2

(1.5.29)

for some ci < oo and all m e N. In view of (1.5.28), there indeed exists a < oo such that for all m e Ν

J um(y)vm(y)dy 0 such that for some m 1 e Ν

.f yeA

, .

. f h(tmy) yeA h(tme)

inf vm(y) - mf — — - > c 4

1. Tauberian theorems

46

for all m > m\. The last two inequalities yield (1.5.29). Let Um{dx)

-

U{dx) =

f(xm)

a{x)dx.

Then by (1.5.29) g(.tmx) _ J_

/(*»)

Ml

Γ g{tmx) Ja

=

Um(A) __}_[(g(tmx)-g(tmy)

Ml

y

f(xm)

Ml JA V

\

f(xm)

)

= p j ^m(^)

- J^(um(y)vm(y)

- um(x)v„(x))

dy^

< Pi ^mW

- jjum(x)vm(y)

- um(x)vm(x))

dyj

=

- um(x) Jj.vm(y)

- vm(x)) dy^

-

(um(A)

Ml

+ c2 suyeA P l"»·^) - M * ) l ·

y

(L5·30)

Therefore, by (1.5.28) and (1.5.30), g(tmx) U(A) hmsup — — - < — — + c 2 um sup sup \vm(y) - t>m(.x)|. m-*oo

J \Xm)

\A\

m—*oo ysA

Passing to the limit as £ 0 in the last inequality and recalling the asymptotic continuity of the sequence of functions {vm(y), m e N}, we obtain gitmx) h m s u p — — - < a(x). m—foo J \Xm) Using lower bounds similar to (1.5.30) (provided that the set A is appropriately changed), we arrive at g(tmx)/f{tmem)

-» a(x),

m ->· oo.

(1.5.31)

By (1.5.23) and the definition of the functions bm(x), 1=

lim I j ^ L = a ( e ) . f(tmem)

(1.5.32)

For χ = e, from (1.5.31) with account for (1.5.32) we obtain g(tme) f(tmem)

j

m-+oo

But if ν > 1, by virtue of (1.5.29) with χ = ve, for m large enough gOmCm) _ Um(em)Vm(em) J \tmem) - um(ve)vm(ve) < um(ve)vm(ve)


1. b / t < 1*1 < 1.

+

f JM(m,y)i ~ Λ « - !1 · t-+ oo αγ ε ( 1 + ε ) If ε in the right-hand side of the obtained inequality tends to zero, we find that [ty]

liminf t-* oo

'

j= ο

a(m, j)ta/

r(t) >

yv/γ.

50

1. Tauberian theorems

Summing over i from [tx (1 — ε)] to [ix], we similarly find that [ty] a

l i m s u p j ) t

yY/γ.

/ r ( t )
0 and choose 8, to in such a way that η \a(m, j ) — a{m,n)\

< ε ^

a(m,k)/n

k=ο

for t > t0 and | j — [/j]) < Sty + 1 where η = [ty]. Then 71

a(m,n)

< a(m,j)

+

s^^a(m,k)/n Jt=0

for all j:[ty(

1 - δ)] < j < [ty] and t > i 0 ; hence for Μ = [ty] - [ty(\ - 5)] we obtain η a(m,n)M


n, I —n = o(n). Then a(m,n) = 0(r (n) / as η

nl+a)

oo and m χ η.

The proof of this theorem repeats the above reasoning word for word. While studying branching processes, we will also use the following Tauberian theorem. Let afunctionrit) be regularly varying at infinity with index γ > —η, a function f (x) be measurable and non-negative in Γ and r-slowly varying at infinity in G (see Definition 1.4.5), and let / ( λ ) < oo for all λ e C. If THEOREM 1 . 5 . 9 .

f(X/t)/t"r(t) as t

ψ(λ) < oo

oo for some solid cone Cq < C and all λ 6 Co, then f(tx)/r(t)

φ(χ) < oo

as t -*• oo for all χ e G. Furthermore, there exists a measure Φ on Γ such that φ is its density in G and Φ(λ) = ψ (λ), VX € C. The proof of this theorem repeats the proof of the third assertion of Theorem 1.3.4 word for word.

1.6. One-dimensional Tauberian theorems Let functions / ( t ) and g(t) be defined and positive for t > a > 0. We write m ~ g ( t )

(1.6.1)

as t —> oo if for any ε there exists to the inequalities (1 - s)g(t( 1 + δ)) < f ( t ) < (1 + *)*(*( 1 - 8)) hold. In the case where (1.6.1) holds, we say that the functions / and g are weakly equivalent at infinity. It is clear that if g(t) is weakly oscillating at infinity (see Definition 1.4.4), then (1.6.1) implies the ordinary equivalence of f ( t ) and g(t) at infinity: f i t ) ~ git),

t

that is, fit)/git) -*• 1 as t oo. First we prove the following Tauberian theorem.

oo,

52

1. Tauberian theorems

THEOREM 1.6.1. Fort > 0, let functions ai(t) > 0, bi(t) > 0, i = 1,2, be given, let a\(t), bi(t) do not increase, and02it), biit) be weakly oscillating at infinity. Let a(t) = ai(t)az(t),

b(t) =

A(t)=

B(t)=

If ait) ~ b{t) as t

[ a(u) du Jo

0+>

bi(t)b2(t), t > 0.

f b(u)du, Jo

and for all λ e ( 0 , 1 ) lim sup B{kt)/B{t) /-»•oo

< 1,

(1.6.2)

then a(t) ~ bit),

oo.

t

This Tauberian theorem, as well as two below, can be referred to as comparison theorems, because they compare the asymptotic behaviour at infinity of two functions and their Laplace transforms. Tauberian theorems of such type were also proved in (Stadtmüller, Trautner, 1979; Stadtmüller, Trautner, 1981; Stadtmüller, 1983; Omey, 1985b; Mikhalin, 1985). The proof of Theorem 1.6.1 is based on the following two lemmas. LEMMA 1.6.1. Let a function f(t) defined for t > 0 be weakly oscillating at infinity and locally integrable on [0, oo). Then the function F(t)=

-fJo/

f(u)du

is weakly oscillating at infinity as well. LEMMA 1.6.2. Let /(/)= Γ g(u)dh(u), Jo where the function g(t) > 0 does not increase, and let the function h(t) > 0 do not decrease and be weakly oscillating at infinity. Then f(t) is weakly oscillating at infinity. PROOF OF LEMMA 1.6.1. For t > Δ > 0 the inequalities 0

^ /ά+Λ

/(")du

~ /o /(") du ^ /;+A

/ο f(u)du
0 such that for t > to Ait) Bit) -

Aikt) Bikt)

-

1 0 and e (0,1) such that for all t >t\,8& (0,

a2it)

1

b2ju)

1+ei'

biict)

< 1 + £l

(1.6.5)

are true, where ει = (1 + ε) 1 ^ 3 — 1. We take an arbitrary S e (0, t\ such that for t >t2 f aiu)du Jet

< (1 +ει)

f Jet

biu)du.

Due to monotonicity of a \ (t) and b\it), hence it follows that for t > t2 aiit)

f a2iu)du Jet

t2 aiit)a2it) 1 +

5 (1 + ε\)

b\ict)b2ict).

hence (1.6.6) for t > t2. Since A(t) ~ Bit) as / —> oo, by the same token for an arbitrary ε2 > 0 there exists $2 e (0,1 — (1 + =

(1 +

Μ

( \ ) ) - ^ g ( X ) ,

0

λ I 0.

(1.6.10)

Then f ( t ) ~ g(t),

t

oo.

R E M A R K 1 . 6 . 1 . I n v i e w o f m o n o t o n i c i t y o f q(t),

f r o m ( 1 . 6 . 9 ) it f o l l o w s t h a t f o r a l l

λ > 0 · rQ&t)

„

0 < hminf

q{t)

,.

— < lim sup —-— < oo, -*oo

t

q{t)

that is, the function q(t) is RO-varying at infinity (see the beginning of the proof of Theorem 1 . 4 . 6 ) . The monotone functions which possess this property are known as dominatedly varying at infinity (Seneta, 1 9 7 6 , A . 3 ) . PROOF. By virtue of (1.6.9) and Theorem A.2 in (Seneta, 1976), there exists η > Μ such that t

um ιπι Um inf — — t-*oo t~HX

f Jι

n + l

g(t)

> 0.

(1.6.11)

k\ung(u)du

For this n, we set a(t)

=

t

n

m ,

b(t)

=

tng(t),

B ( t ) ==

/ ' Jo

b(u)du.

1.6. One-dimensional Tauberian theorems

55

For any fixed c e (0,1), B(t)-B(ct)

fctung{u)du

=

Bit)

^

fiu"g(u)du

gjt)f^u"du

~

ßu"g(u)du'

From (1.6.11) and the last inequalities we obtain .

liminf

f

m - B ( c t )

t-*°o

B{t)

> 0,

which yields (1.6.2). Therefore, by (1.6.10), all hypotheses of Theorem 1.6.1 are satisfied. By virtue of this theorem, f f i t ) ~ t"git),

t -H» oo,

hence we obtain f i t ) ~ git),

oo.

t

The theorem is proved.

•

THEOREM 1.6.3. Let a function git) do not increase, let inequality function f i t ) be differentiate for sufficiently large t, and

f i t ) = Oigit)/t), If there exists Μ such that for any fixed n> dn

-

t

oo.

(1.6.9) hold,

(1.6.12)

Μ

i d "

\

fit) =

o(g(t)).

then as t -*• oo

PROOF. Let η > — ind_ g - 1 (see Remark 1.4.4). By virtue of Theorem 1.5.6, f„gil/t)xt"+1git),

t

> oo,

where T„git) = t"git). Hence, by (1.6.13), ί „ / ( 1 / 0 = oitn+lgit)),

t

oo,

which is equivalent to the relation poo

/ Jo

Since tn+igit)

e~x x" f i t x) dx = oigit)),

—> oo as t =

t -»· oo.

oo), for any b > 0 we see that

ϊ°° e-xx"^^-dx=oi\), )b/t giO Jbl

leta

t —> oo.

56

1. Tauberian theorems

By the theorem of mean value, according to (1.6.12) b > 0 can be chosen so that for t > b the inequalities \ f ( t x ) - f ( t ) \ < o(x

- lk(0,

*>1,

I / O x ) - / ( O l < c o ( l - x)g{tx)/x,

b/t

< χ < 1,

hold, where Co is a constant. By these inequalities, e~xxn(x

| / 2 | < CO

+ j f ' e~Kxn~x

- I )dx

j ^ y d x ^ j ,

where /

2 2

=

r e - * x » !b„

f ( t X )

g(t)

f ( t )

d x •

Hence it follows that Ii = 0(1) as £ ^ oo. Therefore, f°° / e ~ Jb/t

hence we obtain so that for t > b

x

x

nf

(t) ^ - d x = h - I g(t)

f ( t ) — 0(g(t))

as t

2

= 0( 1),

oo. Since

η

t

+ ind_ g

oo,

>

0, b

>

0 can be chosen

f i t ) < cg(t)

(1.6.14)

and gitx)

b/t, f t i x ) = Ο, χ < b/t, for t > b. We check whether the hypotheses of Theorem 1.5.2 are satisfied or not. 1. As t —• oo, xt

—x

>

0,

ftixt)- ftix) = x?fitxt)/git) - xnf{tx)/g(t) =

=

git)

- *") + ^ 7 , i f i git)

t x

0. The hypotheses of Theorem 1.5.2 are thus satisfied, and the proof is complete. • REMARK 1.6.2. As follows from the proof, in order for Theorem 1.6.3 be true it is sufficient that (1.6.12) is true and (1.6.13) is true for some η > — ind_ g. The Tauberian theorem below extends the well-known Tauberian theorem for power series (Feller, 1966, Section XIII.5, Theorem 5). A Tauberian theorem for asymptotic expansions of generating functions is obtained in (Vatutin, 1977c). THEOREM 1.6.4. Let a sequence qn > 0 do not increase and

g(s)

=

Σ 0, and a slowly varying at infinity function

L(t),

q„~na-m-lL{n)/ Γ » .

(1-6.16) then, as η —*• oo,

(1.6.17)

PROOF. F r o m (1.6.16) and Theorem 5 in (Feller, 1966, Section XIII.5) it follows that,

as η

oo, η

~naL{n)/

Γ(α+1),

(1.6.18)

k=0

where k^ η —y oo

= k(k - 1) · · · (k - m + 1). Hence it follows that for any fixed λ € (0,1) as η

J2 k^qk Λ=[λη]+1

~ (1 — λα)ηαL(n)/

Γ(α + 1).

(1.6.19)

1. Tauberian theorems

58

From (1.6.19) and the inequality [Xn]™qn(n-[Xn})
" = * n^ oo n L(n) ~ Γ (a + 1) Γ(α)

(1.6.21)

From (1.6.20) and (1.6.21) it follows that there exists q„nm+l lim a n->°on L(n)

1 Γ (a)'

which is equivalent to (1.6.17). The theorem is thus proved.

•

T H E O R E M 1.6.5. Let functions a(t) and b(t) be measurable and non-negative for t > 0; for λ > 0 let them possess the Laplace transforms α(λ), b(k); let there be s > 0 such that the function b{t) does not increase for t > s and dominatedly varies at infinity:

b(t) lim s u p — — < oo, t—•oo b(2t)

(1.6.22)

1.6. One-dimensional Tauberian theorems

for y > x, y — χ + o(x)

let a(y)

lim sup

-

a(x)

—

χ-κ» If there exists Me

59

< 0.

(1.6.23)

b(x)

Ν such that for any n>

Μ as λ

0

(1.6.24) then a(t)

~

t —* oo.

b(t),

(1.6.25)

PROOF. Since the function b(t) is dominatedly varying at infinity ((1.6.22) holds), by virtue of Theorem A.2 in (Seneta, 1976) there exists M\ > Μ such that for any fixed η >

Mi /0f unb(u)du k m sup — — τ γ γ γ τ — < oo.

(1.6.26)

Hence it follows that for such η and any λ € (0,1) Bn{Xt) SUP

oo ~R~Tt\ on(t)

where

K

(1.6.27)

h

^ B„(t)=

f Jo

u"b(u)du.

It is easily seen, indeed, that (1.6.27) is equivalent to the relation f't u"b(u)

lim sup ^

du

— — > 0,

t->oo

(1.6.28)

fQu"b(u)du

and (1.6.28), in its turn, follows from (1.6.26) because b(t) does not increase. Furthermore, by virtue of Lemma 1.6.2, the function B„(t) is weakly oscillating at infinity (see Definition 1.4.4). Therefore, from Theorem 1.5.4 and relation (1.6.24) it follows that for any fixed n>

Μ An(t)

=

oo,

(l+o(l))Bn(t),

(1.6.29)

where A„(t)

=

f una{u) Jo

du.

We fix an arbitrary λ e (0,1). In view of (1.6.27), there exist ίο, Co > 0 such that for t > to A„(t)

-

A„(tX)

B„(t)

-

B„(tX)

25 and c > 0 such that for Χ > XQ b(x) < cb(2x).

(1.6.31)

We set

From (1.6.23) it follows that there exist xo such that for any χ > χ ι and y e [x, x ( l + Si)] the inequality a(y) - a(x) < sib(x)

(1.6.33)

is true. From (1.6.30) it follows that as χ —*• oo λχ(1+«ι)

/

fX(l+Si)

u"a(u)du

= (1 + 0 ( 1 ) ) /

Jχ

u"b(u)du.

(1.6.34)

Jx

According to (1.6.34), there exists X2 > x\ such that for any χ > X2

/

y

n

a(y)dy>(l-sl)

Jx

y"b(y)dy.

(1.6.35)

Jx

From (1.6.33) and (1.6.35) it follows that for χ > X2 / JX

(a(x)

+

Slb(x))y"dy>

y"a(y)dy> JX

y"b(y)dy, JX

so that («(x) +

n+l

Sib(x))x

η + 1

^

-

η + 1

V

o

+

hence we obtain a(x)

+ eib(x)

>b(x(l

+ Si))(l - e , ) .

By virtue of (1.6.31) and monotonicity of b(x), for χ > X2 the last inequality yields a(x)

+ ceib(x(l

+ δ ι ) ) > a(x)

+ eib(x)

> (1 - e j ) 6 ( * ( l + S i ) ) ,

hence it follows that a(x) > (1 - £ i ) 6 ( x ( l + Si)) - cetb(x(

1 + S,)),

*«».

1.6. One-dimensional Tauberian theorems

61

then e ( * ) > ( l - ( c + l)ei)fc(x(l + Äi))

Vx>x2.

(1.6.36)

By virtue of (1.6.23), there exist x 3 and χ e [y, y/{ 1 - a(x) - exb(y)

> α ( χ ) - ει&(χ(1 - X3 such that for χ > X4 Γ

yna(y)dy X4 Γ y"(a(x) Jx(. 1-SZ)

sib((

1 - 0}, be weakly oscillating at infinity (see Definition 1.4.4). What this means in the case of one variable is that f ( y ) / f ( x ) —• 1 as χ oo, y = x+o(x). By assertion 2 of Theorem 1.4.6, the function f(x) for some s > 0 admits the representation fix) = exp ^η(χ) + J^

(1.6.41)

where the functions η(χ) and ε(χ) are measurable and bounded on [5,00). We fix some / > 0. The Stieltjes transform of the function / ( x ) on R+ denoted / ( λ ) is m

=

f°° f ( x ) dx Jo

(λ + χΥ

(provided that the integral exists for λ > 0). Let us prove the following Tauberian theorem. T H E O R E M 1.6.6. Let a function f (x) be weakly oscillating at infinity, and let g(x) = rix)hix), χ € R+, where r (x) is non-negative and monotone for χ e R+ and the function h(x) is weakly oscillating at infinity. We assume that some representation of f i x ) in the form (1.6.41) obeys the inequalities

- 1 < inf f(x) < sup ε(χ) < / - 1 X2-S

(1.6.42)

x>s

and, as λ -» oo,

|(λ) = (1+0(1))/(λ).

(1.6.43)

iix)

(1.6.44)

Then, as χ —• σο,

= (1 + o ( l ) ) / ( x ) .

Theorem 1.6.6 generalises the corresponding assertions of (Keldysh, 1973; Matsaev, Palant, 1977); see also the book (Kostyuchenko, Sargsyan, 1979, Chapter X). The key difference of Theorem 1.6.6 from the preceding results consists of omitting the requirement that the functions / ( x ) and g(x) must be monotone. Theorem 1.6.6 is proved with the use of some assertions of Sections 1.4 and 1.5. As concerns other results in this field, we point out the book (Pilipovic et al., 1990) and papers (Belogrud, 1974; Sultanaev, 1974; NikolicDespotovic, Pilipovic, 1986; Selander, 1963; Stankovic, 1985b). In order to prove Theorem 1.6.6, we make use of two lemmas given below. L E M M A 1.6.3. Let a function f i x ) be weakly oscillating at infinity, and g(x) = rix)h(x), χ € R+, where r(x) is non-negative and monotone for χ € R + , while the function hix) is weakly oscillating at infinity. We assume that some representation of f i x ) in the form (1.6.41) obeys the inequality

inf ε(χ) > - 1

x>s

and, as λ

0,

£(λ) = ( 1 + ο ( 1 ) ) / ( λ ) . Then, as χ -*• oo,

g(x) = ( l + o ( l ) ) / ( x ) .

1.6. One-dimensional Tauberian theorems

63

Lemma 1.6.3 follows from Theorem 1.5.4. We say that a function a(x) defined in R+ is weakly oscillating at zero if the function a( \/x) is weakly oscillating at infinity. If a function A (Λ:) is weakly oscillating at zero, then from relation (1.6.41) it follows that the representation a{x) = exp ( ω ( χ ) + J P ^^-du^j

(1.6.45)

is true for all χ e (0, ρ] with some ρ > 0, where the functions ft>(x) and ψ(χ) are defined, bounded, and measurable for χ e (0, p]. LEMMA 1.6.4. Leta(x) = xka\{x) > 0, b(x) — xkb\(x) > 0, X E R+, where ai(x) and b\(x) are decreasing functions on R+, k > —I, let thefiinction a(x) be weakly oscillating at zero, and let some representation ofa(x) in the form (1.6.45) obey the inequality sup

< 1.

ο 0 we see that a{X/xm) = Γ Jo

e-^'^aiy)

dy = xma{xm)

Γ Jo

^^-e'^dx. a(xm)

(1.6.49)

We observe that the sequence Jm(X)

_

—

a(xmx)

Γ".

a(Xm)

-V €

K+,

is asymptotically continuous on R + . It is easy to see, indeed, that as m —>• oo, r / \ f , -v Jm(x) ~ Jm(y) because 1

a(xmx) - a(xmy) _ a(xmx) ra(xm) a(xm) a(xmy) a(xmx)

• 0,

m

oo,

Λ _ a(xmy) \ _ V

y

a(xmx)J

χ > 0,

—>· χ > 0

1. Tauberian theorems

64

by the definition of a weakly oscillating function, and, as follows from representation (1.6.45), a(xm)

as m —>· oo provided χ > 0 is fixed. Further, for t < p, there exists a constant c\ < oo such that by (1.6.45) ^ ^ a(t)

= e x p ( ω ( ί χ ) - ω(ί) V

S f W -\cixp, K

+

f Jtx

ν

^-^-dv) J

[ - ' - / J ' '

(1.6.50)

0 < χ < 1,

where a —

s u p (—ψ(ν)), Ο00 5

+ 1)5

If δ tends to zero in the right-hand side of this inequality, taking into account the continuity of / ( x ) we obtain

..hm supb(x m)- < ——

/(l). m-*00 ayx m ) In the same way, with the use of integration from 1 to 1 + δ , we arrive at liminf ^

m-*oo a(xm)

Thus, there exists

> /(I).

b(xm) r = lim - 7 — m^ a(xm)

/(Ο-

g (x),

RecalUng (1.6.51) and the definition of the functions m

we conclude that

/ ( I ) = 1. Thus,

b(Xm) a(xm)

,

— — - —y 1,

m

00.

The last relation contradicts (1.6.48), which proves the lemma.

•

PROOF OF THEOREM 1.6.6. Let us prove that for some ρ < l / s and λ e (0, p ] the transform / ( λ ) admits the representation / ( λ ) = i exp

-γ-dt + ξ(λ)j ,

(1.6.63)

1.6. One-dimensional Tauberian theorems

67

where s(t) is the function from representation (1.6.41) of / ( x ) and £(λ) is bounded for λ e (0,1/p]. We supplement the definition of s(x) and η(χ) with identical zero for χ < s. Since inf ε ( χ ) > — 1, x>s

we see that / ( λ ) -*• oo as λ | 0. Therefore, as λ | 0, / ( λ ) = j f °°

exp ( η ( χ ) + j f *

= (1 + 0(1)) j f ° °

dx +

O(l)

exp ( η ( χ ) + j f * ^y-dt^j

dx.

So, it suffices to obtain a similar representation of j f

exp (τ?(χ) + j f * ^

a ) At.

We observe that

where

In order to prove (1.6.63), it suffices to demonstrate that there exist constants c\, that 0 < ci < /(λ) < c 2 < σο

such

(1.6.64)

for λ € ( 0 , 1 / 4 We represent the integral /(λ) as the sum of the integrals 11 (λ) and / 2 (λ) over u from 0 to 1 and from 1 to oo, respectively: /(λ) = / , ( λ ) + / 2 (λ).

(1.6.65)

For λ € (0,1/s] we obtain /,(X) = j T

e~u exp ^η Q

- jf '

^ d t ^ j d u o

a

= inf ε(ί). '-s

«ρ(-α(-1ηιι))Αι,

68

1. Tauberian theorems

Therefore, for λ € (0, \ /s] Λ (λ) < eC} /

uadu

=

Jo

,

(1.6.66)

α + 1

because a > — 1 by inequalities (1.6.42). Furthermore,

ΛΟΟ

CO

e~" exp(/5 I n u ) d u — eCi J

/

e~uußdu

< oo,

(1.6.67)

where β = supe(i). t>s

In addition, h{^)>ec*

= eC4 f

ί e~uexp(-ß(-]nu))du Jo

OO

/

e~u

/»OO

exp(a In u) du = ec4 J

Jo

e~"ußdu>

e~uuadu

>

0,

0,

(1.6.68)

(1.6.69)

where C4

= inf η (ν). υ>0

Bound (1.6.64) follows from relations (1.6.65)-( 1.6.69). Next, let us demonstrate that / ( λ ) is weakly oscillating at zero. Let μ, λ | 0, /χ/λ —1. Then /(λ) =

e~

X x

f ( x ) dx = ^ f

= ( l + o ( l ) ) J " e - »

u

o

° ° e - ^ f ( £ u ) du

f ^ u ) d u

= (1 + *(!)) ( / V ) + j f * - " / < « ) ( l -

*,) .

So, it suffices to show that, as λ, μ | 0, λ / μ —»· 1, - ^

f

j

du

=

d·6·70)

First, we see that, as λ, μ | 0, λ / μ —> 1, for any fixed t > 0

jf

(

'

j

f

= O(l) = 0 ( / < r t ) .

- jf ·•*· / ( r ) (1.6.71)

1.6. One-dimensional Tauberian theorems

69

We fix an arbitrary ε > 0, and for this ε choose t > 0 and 8 > 0 in such a way that the inequality ,

/(ftp < ε

(1.6.72)

holds true for u > t and | μ / λ - 1| < 8. Then (1.6.71) and (1.6.72) yield lim sup J — [ Γ e - » μ4·0,λ/μ-*1 / ( μ ) J JO

u

f ( u ) \ \ - ^ j ^ \ d u = < ε. \ /(«) /

Since ε is an arbitrary positive number, (1.6.70) now follows from the last inequality. As we know (Seneta, 1976, Theorem 2.5), a Stieltjes transform is a double Laplace transform, namely, for λ > 0 pOO ΛΟΟ /(λ) =

/

a(x)e~xkdx,

g(X) =

Jo

b{x)e~xXdx,

/

(1.6.73)

JO

where a(x) = x ' - ' / M / Γ(/),

&(*) = jc'-'^CJC)/ Γ(/).

(1.6.74)

As we have seen, / ( x ) weakly oscillates at zero, hence so does the function a(x), and from (1.6.63) it now follows that for χ e (0, l/p] a(x)

= x'~2

exp ^

= χ1'2

exp

'

^y-dt

+ ξ(χ)

( J * ^ ^ d u

= exp

du +

- In Γ(/) j

+ ξ(χ)

ω(χ)^

- In Γ(/)^

,

where f ( x ) = 2-1

+ ε(1/χ),

ω(χ) = ξ (χ) - 1η Γ(/) + (2 - /) 1η ρ.

By (1.6.42), sup ψ(χ) = 2 — 1+ 0 O f o r x e G = intT; (2) for any family Uk,k € I , of linear operators of first (second, third) type which leave invariant the cone Γ, there exist x° € G and a subsequence {Ukm, km -> +00} as m -> + 0 0 such that ßiU^x) ,,,

, 0\

„

s

>0,

g(x)

χ € G,

as m —• + 0 0 uniformly with respect to χ in an arbitrary compact Q C G, where g(x) is a continuous function for χ e G; (3) there exists mo such that ß(Ukmx)

< ψ(χ),

m > mo,

and there exists q such that

I

ψ(χ) 1 + |*|
0 there exists δ > 0 such that | / ( x ) - 1| < ε for ΔΓ(Χ) > 5,

/ ( χ ) - » • 1,

χ o o

in Γ.

The following Tauberian comparison theorem is true (see (Drozhzhinov, Zavyalov, 1984) or (Vladimirov et al., 1988, Section Π.6.2, Theorem 2)). THEOREM 1.7.2. Let μ and ν be non-negative tempered measures on Γ. (1) If μ is an admissible measure of first type for the cone Γ and v(y)/fl(y) y -*• 0, y e C = int Γ*, then ν ( χ ) / μ ( χ ) —> 1,

χ

oo in Γ.

—• 1 as

1. Tauberian theorems

72

(2) If μ is an admissible measure of second type for the cone Γ and for any b > 0 v(y)/pL(y)^lforAc(y)^0,

yeC,

\y\ < b,

then for any δ > 0 ν(χ)/μ(χ)

-*• 1 for |x| -> +oo,

ΔΓ(Χ) > δ.

(3) If μ is an admissible measure of third type for the cone Γ and v(y)Wiy)

I for AcOO - » ο ,

yeC,

then ν(χ)/μ(χ)

->• 1 for |x|

+oo,

χ e G.

Let, as before, {Uk, k e 1} be some family of non-singular linear operators which leave invariant the cone Γ, and let Ak, Xk be defined by relation (1.7.1) DEFINITION 1.7.3. A measure μ on Γ is said to be completely admissible for a family {Uk, k 6 / } if the following conditions are satisfied: (1) μ(χ) > Ofor χ eG = intT; (2) there exists a vector x° € G such that ß(Ukx)

g(x) > 0,

k

+oo,

k € I,

uniformly with respect to χ in any compact Q C G, where g(x) is a continuous function in G\ (3) there exists

such that ß(Ukx) ß(Ukx°)

< ψ(χ),

k > ko,

χ e G,

where ψ(χ) is a tempered function on Γ, that is, relation (1.7.2) holds. In (Vladimirov et ai, 1988, Section II.6.1, Theorem 3), the following Tauberian comparison theorem is proved. THEOREM 1 . 7 . 3 . Let μ and ν be non-negative

tempered

measure μ be completely admissible for a family {Uk,k are satisfied:

measures

on Γ, and let the

e I}. If the following conditions

(1) there exists an open set Ω C C = int Γ* such that v(VkkJy) ' fi(Vky)

• 1,

k

co,

k e I,

y e Ω,

Vk = (£/£)

;

1.7. Tauberian theorems of Drozhzhinov-Zavyalov type

73

(2) there exist numbers M, ß, ko, and a vector e € C such that 0• oo,

k e I,

uniformly in χ e Κ for any compact Κ C G = int Γ. DEFINITION 1.7.4. A measure Μ on Γ is said to be a-admissible for a cone Γ, if for any family of non-singular linear operators {Uk,k € 1} which leave invariant the cone Γ such that λ*. > pA% there exists a subsequence {Ukm, m —* + o o , km e 1} for which μ is completely admissible. Let Da(T) stand for the set of all α-admissible measures for a cone Γ. It is clear that a < 1. THEOREM 1.7.4. Let a measure μ satisfy the hypotheses of Theorem 1.7.1. Then μ € Da(Γ) for any a < 1. In (Drozhzhinov, Zavyalov, 1990), the following Tauberian comparison theorem is proved. THEOREM 1.7.5. Let μ and υ be non-negative tempered measures on Γ, and μ € Da(T). If for an arbitrary constant c ι > 0 vM

lforAc(y)^0,

\y\ < c^^y),

ysC,

then for an arbitrary constant cz > 0

The appropriately altered Theorems 1.7.3 and 1.7.5 are true in the case where one of the measures is complex-valued (Drozhzhinov, Zavyalov, 1992, Theorems 6 and 7). In (Drozhzhinov, Zavyalov, 1990, Theorem 4), the following Abelian comparison theorem is given. THEOREM 1.7.6. Let μ and υ be non-negative tempered measures on Γ, and μ 6 Da{T). If for an arbitrary constant c > 0 v(x) μ(χ)

-*• I for I JE I -> OO,

ΔΓ(Χ) > c\x\a,

xeG,

\y\ 0 v(y)

^lforA

c

(y)^0,

74

1. Tauberian theorems

In (Drozhzhinov, Zavyalov, 1990, Theorem 3), an Abelian comparison theorem for tempered distributions was proved. The proof of that Abelian theorem came up against severe analytical difficulties. Additional constraints should be imposed on the functions under consideration naturally referred to as Abelian conditions. The corresponding quite complicated counterexample was considered there (Section 3). In what follows, we consider a much more wide class of cones for which Tauberian theorems 1.7.2,1.7.3, and 1.7.5 are true, as well as Abelian theorem 1.7.6. DEFINITION 1.7.5. A convex cone Γ is said to be homogeneous if for any vectors a and b in G — int Γ there exists a non-singular linear operator U which leaves invariant the cone Γ such that Ua = b.

Let Γ be a closed convex acute solid cone in R" with apex at zero (see the beginning of Section 1.1). As before, let Γ* denote the cone dual to the cone Γ: Γ* = {y: .y e R", (>\ x) > 0 Vx e Γ}, C = int Γ*. Let 5"(Γ) stand for the space of tempered distributions with supports in Γ; it is dual to the space 5(Γ) of infinitely differentiable functions φ(χ) such that pm{ ( d u ) - ( - 1 )mm\ cm + Jo (4)

tmr(t)

A (a + m) Γ(α)

+ o(l)

ast->oo, μ((ί, oo)) = r(t)

_αΓ(α)

+

od)]·

REMARK 1.7.1. The implication 1 => 4 is proved in (Nevels, 1974), see also Problem 15 in Chapter XVII of (Feller, 1966). REMARK 1.7.2. In (Drozhzhinov, Zavyalov, 1995a), a Tauberian theorem for asymptotic expansions of Laplace transforms of measures concentrated in the positive octant is also proved. Let us present a Tauberian theorem of Drozhzhinov-Zavyalov type for characteristic functions (Drozhzhinov, Zavyalov, 1995b)). Let f ( x ) be a characteristic function of some random variable with distribution function F(x): +oo

/

e"xdF(t), •OO


0, we set • oo and k(k) that A (k)

->ooask-+oo,kef,

as k —*• oo, ä: e

{Uk,

k E / } is a family of first type if

->· oo as k -*• oo, k e /; of second type if there exists b > 0 such and λ(Α:) > b > 0, k e /; of third type if A(k)

-y

oo

I.

W e introduce functions which regularly vary in G along the family U. DEFINITION 1.8.2. W e say that a function / ( x ) , which is defined, positive, and measurable in G, is regularly varying in G along a family U f € R(U,

G), if for some vector e e G and all χ G G as f(Ukxk)

J:iT

f ( U

J k

0,

e )

φ(χ)

— {Uk,

k

€

/ } and write

—> x, k —• oo, k € I,

< oo.

(1.8.3)

From (1.8.1) and Theorem 1.4.2 it follows that ί

^

f { U

for any compact Κ write ψ — He(U,

^ φ ( χ )

Σ k

e )

k^

6 ( 0 , oo),

kel,

oo,

(1.8.4)

y

C G, while φ(χ)

is continuous in G.

In accordance with (1.8.3) w e

G).

R E M A R K 1.8.1. The set R(U,G)

does not depend on the vector e e G.

Further, if

(1.8.3) holds for some vector e e G, then it holds for all vectors e\ e G, and the function φ is multiplied by um

k-nx,

km

f ( U

m*. k

e i )

DEFINITION 1.8.3. A function f ( x ) defined in G is said to be completely admissible for a family of operators U =

{Uk, k € / } , if / € R(U,

G),

f is locally summable in G

and there exists ko such that

4t^4 ko,

e )

kel,

χ 6 G,

(1.8.5)

where e is a fixed vector in G and η is a tempered function in G :

L Jo

η(χ)

1 +

< oo kl9

for some q. REMARK 1.8.2. Definition 1.8.3 is equivalent to the definition of a O-completely admissible function for a family U = Section 5.2).

{Uk,

k € 1} in (Vladimirov et al., 1988, Chapter II,

1. Tauberian theorems

78

DEFINITION 1.8.4. A function f ( x ) defined in G, is said to be admissible for a cone Γ, if for an arbitrary family of linear operators U — {Uk, k € / } which leave invariant the cone Γ there exists a subfamily V = {Uk, k e J ζ 1}(J is unbounded) for f ( x ) is completely admissible. Let D(T) denote the set of all admissible for the cone Γ functions. DEFINITION 1 . 8 . 5 . A function f ( x ) defined in G is said to be admissible of type m, m = 1,2,3, for a cone Γ if for an arbitrary family of linear operators of type m U = {Uk, k € 1} which leave invariant the cone Γ there exists a subfamily V = {Uk, k e J c / } (7 is unbounded) for f(x) is completely admissible.

Let D m ( r ) , m = 1,2,3, denote the set of all admissible functions of type m for a cone Γ. Let R m ( r ) , m — 1,2,3, denote the set of all functions f ( x ) defined in G such that for an arbitrary family of linear operators of type m U = {Uk, k e 1} which leave invariant the cone Γ there exists a subfamily V = {Uk, k e J C / } (J is unbounded) such that / e R(V, Γ) (see Definition 1.8.2). Let Ε ( Γ ) denote the set of all functions f ( x ) defined in G such that for any family of linear operators U = {Uk, k € 1} which leave invariant the cone Γ there exists a subfamily V = {Uk, k e J c / } (J is unbounded) such that / € R(V, Γ) (see Definition 1.8.2). It is clear that Β>(Γ) c ϋ 3 ( Γ ) c ϋ 2 ( Γ ) c Οι(Γ) and Κ(Γ) C R 3 ( D C R 2 ( D C κ , ( Γ ) . Let us give examples of functions of R ( r ) and Β(Γ). Let R ^ = {χ =

€ R " , X/ > 0 VI =

1,...,«}

be the positive coordinate octant, V

n = J x - ( x o , X l , . . . , X n ) € R n + 1 , X0 >

+ ··· + * ί |

be the future light cone in R" + 1 . E X A M P L E 1.8.1. Let f which satisfies the relation

a