Convolution powers of spread-out probabilities

A NNALES DE L ’I. H. P., SECTION B

M ^ICHAEL L ^IN

R ^AINER W ^ITTMANN

Convolution powers of spread-out probabilities

Annales de l’I. H. P., section B, tome 32, n

^o

5 (1996), p. 661-667

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Convolution _powers of spread-out probabilities

Michael LIN 1

Rainer

WITTMANN 2

Ben-Gurion University ^{of the} Negev, Beer-Sheva, ^Israel

Institut fur Mathematische Stochastik, Lotzestrasse 13, Gottingen, Germany

Vol. 32, ^nO 5, 1996, p. 661-667 Probabilités et Statistiques

ABSTRACT. - Let G be a

locally compact a-compact

group, and

let J-L

^be

a

spread-out probability, adapted

^and

strictly aperiodic.

We prove that for any continuous isometric

representation T (t)

^{in a}

uniformly

^{convex Banach}

space, ~~ - ~ ⁽

^{-~ 0}

^{(where U~}

Soit G un groupe localement

compact

denombrable a

l’infini,

et soit _tc une

probabilite ^etalee, adaptee

^et strictement

aperiodique.

^Nous

prouvons que pour ^toute

representation

^continue

T (t)

^{par isometries}

d’un espace de Banach uniformement

(

^{-~ 0}

^(ou U~

1. INTRODUCTION

Let G be a

locally compact a-compact

group with

right

^{Haar measure}

A. For a

regular probability tc

^on

G,

the convolution

operator f (t)

⁼⁼

1 Research partially supported by the Israel Ministry ^{of Science.}

2 Heisenberg fellow of the Deutsche Forschungsgemeinschaft.

Annales de l’[nstitut Henri Poincaré - Probabilités et Statistiques - ^0246-0203

Vol. 32/96/05/$ 4.00/@ Gauthier-Villars

662 M. LIN AND R. WITTMANN

j

^{is a Markov}

^opeartor

^{with a-finite invariant measure, which}

is the tc-average of the translation

operators 03B4s* f (t)

⁼

f (ts).

Let S be the

support

^{of the}

probability

tc. We say

that

^is

adapted

^{if the}

closed

subgroup generated by

^{S is}

^G,

^and

strictly aperiodic

^{if the smallest}

closed normal

subgroup,

^{a class of which contains}

^S,

^{is G.}

An

important property

^{for the}

study

^{of the}

asymptotic

^{behaviour of}

is the

ergodicity

^of tc,

i.e., that ( ~ ^~~-1

^*

^],

^{-~ 0 for every}

f E L1 (G, A) with J ^fdA

^{= 0.}

^Ergodic probabilities

^are

necessarily adapted [A].

In

applications,

^{we often}

have tc spread-out (i.e.,

^{for some n >}

0,

^is

not

singular

^with

respect

^to

A).

^Glasner

[G] proved

^that

if tc

^{is an}

ergodic

^and

strictly aperiodic spread-out probability

^on

^G,

^then

is

compact

^{we even}

have

^{--~ 0}

^[M];

^see

^[RX]

^{for more}

^results.)

If

]

^--~

^0,

then for every bounded continuous

representation T(t)

^{in a Banach}

]

^-~

^0,

^where

is the -average of the

representation.

Glasner also _gave an

example

^for tc

adapted, strictly aperiodic

^and

spread-

out,

with

^{= 2 for any}

^{n, k}

^{> 0.}

^Following

^Jaworski

^[J],

let _r~

= 2 (tc

⁺

^{with p}

of Glasner’s

example. Clearly also 1J

^is

adapted

^and

strictly aperiodic, ]

^{--~ 0}

by [F]. However,

I

⁼ 2 for every ^n, since all the powers

of

^are

mutually singular. (See [LW]

^{for related}

results.) Nevertheless,

^{it was shown in}

[DL]

that

if tc

^is

adapted, strictly aperiodic

^and

spread-out,

^then for any continuous

representation by

isometries in a

uniformly

^convex Banach space, the iterates of the -average

Up

^converge

strongly (necessarily

^{to a}

projection

on the common fixed

points).

^{In this} paper ^we

improve

^this

result, by showing

that in fact

]

^{-~ ~ .}

2. OPERATOR-NORM CONVERGENCE IN UNIFORMLY CONVEX SPACES

PROPOSITION 2.1. -

Let p be a spread-out probability

^{on a}

locally

compact

03C3-compact

group. Then

for

^{every ~ > 0} there exist an

integer

^{N and}

neighbourhood

^A

of

^e, such that

for

^{n > N and we have}

~03B4t* n - 03B4s * n| ( ~, and ~T(t)Un - T(s)Uu ~ ~ for any

contractive continuous

representation.

Proof -

^Let

tcn

^{= vn}

-~ r~~

^{be the}

Lebsegue decomposition

^of

since ,

^is

spread-out,

^{0 for some no, so 1. Hence}

I I

Annales de l’lnstitut Henri Poincaré - Probabilités ^et Statistiques

Fix e > 0. There exists N

] e/3.

^{Since vN «}

^A, by continuity

of the translations in

£1 ( G, À)

there exists ^a

neighbourhood

^A

of e such that

~~bt *

^{vN -}

] e/3

^{for t E A.}

For

n >

^{N and}

t-1s

E A we now have

For a contractive

representation,

THEOREM 2.2. -

Let be a spread-out adapted

^and

strictly aperiodic probability

^{on a}

locally compact a-compact

group G. Then

for

every continuous

representation of

^G

by

isometries in a

uniformly

^{convex Banach}

space, ^we have

i ~ U~ + 1 - U~ ~ ~ ^]

^{-~ 0.}

Proof. -

We may ^assume

T ( e )

⁼

^7,

^{so all}

T(t)

^are invertible. We denote

U~ by

^{U. Since} the theorem is obvious if Un ⁼ 0 for some n, ^{we assume} that 0 for _every n.

Let am be a sequence of natural numbers

increasing

^{to oo, with}

m T

^o0

(e.g.,

^{= Let 0 1 with}

T

¹

slowly enough

^{to have}

0

(e. g.,

^{= 1 - for =}

Fix m with m > 303B1n, and define Xm = {p e X : Um-203B1mx~0}.

For x e

X~

^{we have}

!7~~c /

⁰

for j

^{m - so we can define}

Clearly D ( m, x )

^1.

For x ~

^{0 we define}

i( m, x)

^{as follows:}

(i)

^{If x E}

Xm

^and

D(m, x) :S

^{1m, then}

i (m, x)

^{= m -}

(ii)

^{If x E}

Xm

^and

D(m,x)

^{> 1m’ let}

i(m,x)

^{= min}

(iii) i(m, x)

^{= m - am for}

Xm.

Let

{x

^E

^{Xm : 1,} D(m,x) :S ~ym ~.

^{For x E}

^A~,

^{we have}

m -

3am

^{+ 1}

inequalities

Vol. 32, nO 5-1996.

664 M. LIN AND R. WITTMANN

Starting with j

^{= m -}

203B1m

^and

iterating

^back

(with jumps

^of

2am)

^we

use 2 m ~ -1 inequalities

^{to obtain}

Let

Bm

⁼

{~

^e

Xm : 1, D(w,a;)

^>

~}.

CLAIM. - Let t E

Sk,

^{where S =} supp . For ^{m >}

3am

^let

Then

limm~~ 03B4k(t, m)

^{= 0.}

Proof. -

^Fix p ^> 0.

By

^uniform

convexity,

there exists 1 > e >

0,

such that

1 ^{C 1,}

^>

^1,

⁺

2(1 - é) imply I

^p.

By Proposition 2.1,

^{there exist}

N,

^{and a}

neighbourhood

^{A of} e, such that

E A ~ ~ (

^{~ for n >} N. Define V ⁼ tA.

Since t E > 0.

_

There exists mo such that for

m ~

^{mo, we have}

(i) (3m

where ⁼ 1 - _,m.

(ii) 2 a’.,.t >

^N.

⁽ⁱⁱⁱ⁾

^k.

^(iv)

^{m >}

Fix m > mo. Let x E

Bm.

^Denote

i(m,x) by i,

^{since x and m}

are now fixed. Then 03B1m i m -

203B1m by definition,

^{and satisfies} Since k _am,

for j

^{cx~ we have}

Hence, for j

The

integrand (and

^{hence the}

integral)

is bounded above

by

^We

show that for

some sj ~ V ( j

^{we have}

Indeed,

if not, ^we

obtain, by integrating

^{over V and over}

V~,

and the strict

inequality yields

^a contradiction.

Annales de l ’lnstitut Henri Poincaré - Probabilités et Statistiques

Hence,

^for

fixed j

^{with we have}

since sj ^{E tA,} and j

^{> > N. Hence}

since

13m

By

the uniform

convexity

^choice

of e,

This

yields 03B4k(t, m)

^{p for}

m >

^mo, which proves the claim.

k [m 203B1m]-1

Proof of

the Theorem. - Fix t E

Sk.

Let

03B2k (t, m)

^--

maX{03B3[m 203B1m]-103B3m

^{J_ ,}

b~ (t, m) ~

^so

^,~~ (t, m)

^{-~ 0}

by

the claim.

m-> o0

Let

t,

^{s E}

S~,

^{and fix m with}

3

^{> 21~. Then}

Taking j

^{= in}

(2), and j

⁼

^{am - k}

ⁱⁿ

(3) (since a~.,.L - 1~ >

we obtain for any ^{x E}

Bm

Since cxm ^{= m} for x e

Am,

^we obtain from

(1)

^that

(4)

^and

(5)

hold for x e

Xm with ~x~ ]

^{1. Since = 0 for}

x ft ^{X m}

^and

Vol. 32, n° 5-1996.

666 M. LIN AND R. WITTMANN

+ a"Z ⁼ rra, we conclude that

(4)

^and

(5)

hold for every ^{x E} X

with 1.

From

limm-+CX) m)

^{= 0 it now} follows that U is contained in

We show that G’ is a closed

subgroup.

^{It is}

trivially

^{closed under}

inversion. If

s, t

^E

G’,

^{then st e}

G’

since

holds

for j

⁼

i(m,x)

^{+ am.}

We show that

G’

is closed. Let

to E G’. By Proposition 2.1,

^{for e > 0} there exist N and a

neighbourhood A

^of e, such that

for j >

^{N and}

t-1s

E A we

have (

^{e. Let}

^t’

^{E G’ be in}

toA. Then,

since j (m, x)

> 03B1m and

t-10t1

^E

^A,

^for

sufficiently large

^{m we have}

Hence

Since c > 0 was

arbitrary, to E G’,

^so

^G’

^{is a closed}

subgroup. By

strict

aperiodicity, ^G’

^{= G.}

Define

fm(t)

^{= Then}

_fm(t)

^{-~ 0}

everywhere

^{on G.}

Strong continuity

^{of the}

representation yields

^that

f m ( t)

is lower

semi-continuous,

^{so is} Borel measurable.

By Lebesgue’s theorem,

0.

Fix c >

0,

and let mo be such

that f

^{~ for m > mo. For}

such _m, we obtain for

every ^1, (since j(m,x)

⁼

i(m,x) ^{-~- ~m}

^rrz

by construction),

^that

Annales de l’lnstitut Henri Poincaré - Probabilités et Statistiques

REFERENCES

[A] ^R. AZENCOTT, Espaces ^{de Poisson des} groupes localement compactes, Springer

Lecture Notes Math., ^Vol. 148, ^1970.

[DL] ^Y. DERRIENNIC and M. LIN, Convergence ^of iterates of averages of certain operator representations ^{and of} convolution powers, J. Funct. Anal., Vol. 85, 1989, pp. 86-102.

[F] S. R. FOGUEL, Iterates of a convolution on a non-Abelian group, Ann. Inst. Poincaré, Vol. B11, 1975, pp. 199-202.

[G] S. GLASNER, On Choquet-Deny measures, Ann. Inst. Poincaré, Vol. B12, 1976, pp. 1-10.

[J] W. JAWORSKI, ^{On the} asymptotic and invariant 03C3-algebras ^of random walks

on locally compact groups, Prob. Th. ^and Related Fields, ^Vol. 101, 1995,

pp. 147-171.

[LW] M. LIN and R. WITTMANN, Convergence ^of representation averages and of convolution _powers, Israel J. Math., Vol. 88, 1994, _pp. 125-157.

[M] ^D. MINDLIN, Rate of convergence for the convolutions of measures on compact groups, Theory of Probability ^and Appl., ^Vol. 35, 1990, _pp. 368-373.

[RX] ^{K. A. Ross} and D. Xu, Norm convergence of random walks on compact hypergroups, ^Math. Z., Vol. 214, 1993, pp. 415-423.

(Manuscript accepted May 30, 1995. )

Vol. 32, n° 5-1996.