Strong laws of large numbers for weighted sums of random elements in normed linear spaces

(1)

STRONG LAWS OF LARGE NUMBERS FOR

WEIGHTED

SUMS OF

RANDOM

ELEMENTS

IN NORMED LINEAR

SPACES

ANDREADLER

Department

ofMathematics Illinois Institute ofTechnology

Chicago, Illinois 60616U.S.A.

ANDREW ROSALSKY

Department

ofStatistics

Universityof Florida

Gainesville,Florida32611 U.S.A.

ROBERT L. TAYLOR

Department

of Statistics

Universityof Georgia

Athens, Georgia 30602

U.S.A.

(Received December 16, 1988 and in revised form February 20, 1989)

ABSTRACT.

Consider asequence of independentrandomelements

{Vn,

n

>

in a real separable

normed linear

space

(assumedtobe a Banachspaceinmostof theresults),andsequencesof con-stants

{a,,

n

>

and

{ha,

n with0

<

b, "["

oo.

Sets of

conditions areprovidedfor

{an(V

EVn)

n

>

toobeyageneralstrong law oflarge numbersof the form

aj(Vj

EVj)/b

n--> 0 almost certainly. Thehypotheses involve the distributions of the j=l

{V,,

n

>

},thegrowthbehaviorsof

{a

n

>

and

{bn,

n

>

},and for some of the results

imposeageometricconditiononX.

Moreover,

Feller’sclassicalresultgeneralizing

Marcinkiewiez-Zygmundstrong lawof largenumbers is showntohold for randomelementsin a real

separable

Rademacher typep

(1

<

p

<

2)Banachspace.

KEY

WORDS

AND PHRASES.

Real separableBanach

space,

independentrandom elements, normedweightedsums,stronglaw

of

largenumbers, almost certain

convergence,

stochastically

dominatedrandom elements,Rademachertypep, Beck-convex normedlinear

space,

Schauderbasis.

(uniformly) tight sequence.

(2)

508 A. ADLER, A. ROSALSKY AND R.L. TAYLOR

1.

INTRODUCTION.

Let (fl,

F, P)beaprobability spaceand letX be a tealseparable normedlinearspacewith norm

.

I. It

issupposedthat

X

isequippedwith itsBorelo-algebra8. Thatis,8 is theo-algebra

generatedbythe class ofopensubsets of

X

determinedby I.

II. A

random

element, Vin

x

isan /:-measurabletransformationfrom fltothemeasurable

space

(X,8). Theexpectedvalue of

V,

denoted

EV,

isdefined tobe the Pettisintegral provided itexists. That is,Vhasexpectedvalue

EVE X

iff(EV) E(f(V))foreveryf

X*

where

X*

denotes the (dual)spaceof all continuous linear functionals on X. The definitions ofindependenceandidenticallydistributed for random ele-mentsaresimilarto thoseinthe(real-valued)random variable case.

Considera

sequence

ofindependentrandomelements

{Vn,

n

>

},

allofwhose

expected

values exist. Let

{a

n

>

and {b n

>

beconstantswith0

<

b

"1"

**. Then

{an(V

EVn),

n _> is saidto obeythegeneral strong law oflarge numbers (SLLN)withnorming constants

Ib

n,n

>

1}

ifthenormed weightedsum

E

aj(Vi

EVj)/bn

converges

almostcertainly to

the zero element in

X

(denoted by 0),and this will bewritten

--,

0a.c. (1.1)

bn

Herein, the main resultsfurnishconditions on X,onthe distributions of the

{Vn,

n

> 1},

and onthe

growthbehavior of the constants

{a

n,n

>

and

{bn,

n >_ which ensurethat the

SLLN

(1.1)

obtains.

In

mostof the resultsX isassumedtobe aBanachspace,and inmanyof the results

{V

n

>

is assumedtobe stochasticallydominatedbyarandom element

V

inthe sense that for

someconstant

D <

P{

IV,,I

> t} < DP{

IDVI

> t},

>

0,n

>

1.

(1.2)

Of course,

(1.2)

isautomaticwith

V

and

D

if the

{V

n,n areindependent and

identi-cally distributed (i.i.d.)and even in this case theresultsare new.

In

Section 3,

SLLN’s

are estab-lished under

geometric

conditions on Xwhereas in Section4,

SLLN’s

areestablishedwithout such conditions. The

SLLN

problemwasstudiedbyAdler andRosalsky [1, 2]in the random variable case, andsomeof those results will be extendedtothe random element casein thecurrent work.

Taylor

[3] provided

a

comprehensive

andunified treatment of results under whose conditions

anjVj

---> 0a.c. where

{V

n,n >_ areindependent, mean zero random elements in a real

(3)

argumentsin_{Taylor’s monograph utilized}_{a result of}_Rohatgi

[4]

whichwill_{now be stated.} (Rohatgi’s_{work generalized earlier work of Pruitt}

[5].)

THEOREM

(Rohatgi [4]).

Let

{Xn,

n

I]

be

independent,

mean_{zero random variables and} let

X

be an

Lp

_{random variable for some}

p

>

_1.

Suppose

_that

Xn,

n

>

1}

is

stochastically

dom-inatedby

X

in thesense that

PI

IXnl

>

t} -< P{

IXl

>

t], t>0,n>l.

Let

{anj

<

_j

<

n, n beconstants

satisfying

_{lira anj} 0 foreach

>

lanjl

O(1),

_and

j=l

max

a,

jl

O(n-1/(1>-1)).

l<j_<n Then

anjXj

-->

0a.c.

j--I

In

Theorems8 and9andCorollary 2 of the currentwork,versionsof some of the results

(1.3)

pr6sented

inTaylor[3] willbe obtained underlessrestrictive conditions butonlyforthe case where

anj

aj/b

n,

<

j

<

n,n

>

1,where

{an,

n

>

and

{b,,

n

>

areconstants. The_argumentswill notinvolveRohatgi’s theorem but, rather,willemploy

Corollary

below.

Corollary

plays

arole

intheproofssimilartothe role thatRohatgi’stheoremplayedinestablishingthe counterparts

presentedinTaylor[3]. Corollary has lessstringentconditions thanRohatgi’stheorem when

anj

aj/b,,

<

n,n

>

1. Specifically,ifthat choice of

{an

j,

<

n, n satisfies(1.3),then

an

O(n

-u(p-l))

(1.4)

bn

which is stronger than the condition

O(n

-/p)

(1.5)

bn

of

Corollary

1. Thus,if

{an

j,

<

n,n satisfies

brian

O(n

a)

forsome 1/2

<

t

< I,

thento invokeRohatgi’stheoremrequiresthat(1.4) and themomentcondition

E

IX

p

<

_{hold where} p

>

+

__1

>

2, whereastoinvoke

Corollary

merely requires that (1.5) and the(weaker)moment

1

condition

E

IX

P

<

_{hold where}₂

>

_p

---"

_{_> 1.}

For

_example,_fori.i.d,random variables

IX

n

>

with

EX1

0,the classical

Ko.lmogorov

SLLN

Xj/n

-

0 a.c.follows from

Corollary

butnotfromRohatgi’stheorem (which wouldrequire

EX

2< 00).

(4)

510 A. ADLER, A. ROSALSKY AND R.L. TAYLOR

Forrandomelementsin a realseparableBanachspace, the study ofthe

SLLN problem

dates

backtothepioneeringwork by Mourier[7] (seealso Laha and Rohatgi[8, p. 4:52]orTaylor [3, p. 72])wherein a directanalogue ofthe classicalKolmogorov

SLLN

was established.

More

precisely,

Mouriershowed thatff

IV

n

>

arei.i.d, randomelements in a realseparableBanachspaceand ifEl

IVtl

<

*,,,then

(Vj

EVI)/n

--> 0a.c. (Anewproofof Mourier’sSLLNhasrecently been

j=l

discoveredby Cuestaand

Matran [9].) For

randomvariables,theKolmogorov

SLLN

was

general-izedbytheMarcinkiewicz-Zygmund SLLN (see, e.g.,Chow and Teicher [10, p. 122])which, in turn, was generalizedbyFeller[11].

A

randomelement version of Feller’s result ispresentedin Theorem 4 below wherein it is assumed that the Banach

space

isof Rademachertypep(I

<

p

<

2).

2.

PRELIMINARIES.

Some

definitions will be discussed andlemmaswillbepresentedpriortoestablishingthe main

results.

Let

{Yn,

n

>

be a Bernoullisequence,that is,

{Yn,

n arei.i.d,_{random variables with}

P{YI

-I 1/2. Let X bearealseparableBanachspaceand let

X

x

XxX x

and define

C(X)

{v

n

>

1}

X**"

Ynvn

converges

inprobability

n-1

Let

<

p

<

2. Then

X

is saidto beofRademachertyp_epifthere exists aconstant0

< C <

such that

El

l,

Y.

nvnlIP_<CEIIVnllP

n=l n=l

for all

{vn,

n

>

C(X).

Hoffmann-Jdrgensen

and Pisier

[12]

provedfor

<

p 2 thatareal

separableBanachspaceisofRademachertypepiffthere exists a constant0

< C <

such that

El

z_,Vjl

It’ <

CT_.,

El

IVil

It’

foreveryfinite collection

{Vt

V ofindependentrandomelementswith

EVj

0, El

IVjl

It <

,,,,

<jn.

Ifa realseparableBanachspaceis ofRademachertype

p

for some

<

p

<

2,then it is of Rademachertypeqfor all

<

_q

<

p.

Every

realseparableBanachspaceisofRademachertype (at

least) whilethe

Lp-spaces

and

-spaces

are of Rademachertypemin 2,p forp 1.

Every

real

separableHilbertspaceand real separablefmite-dirnensional.Banach space isof Rademacher type 2.

(5)

I1+v

__.

:!:

VNI

< N(1-E)

for somechoiceof+and signs. This propertyhas beenextensively studiedby Giesy [13].

A

real

separableBanach spaceisBeck convex iff it is of_{Rademacher type p for some p >}_1.

A

_Schauder basis for a normed linear

space

is a

sequence

{1

i,

>

1}

c g suchthat foreach

v

X

thereexists aunique sequence ofscalars

{t

i,

1}

suchthat

rn

lim

tii=v.

(2.1)

rn--***_i--I

A

sequenceof linear functionals

{f

i,

>

(calledcoordinatefunctional$for the basis

{i,

>

})

canbedefinedbyletting

fi(v)

ti, :> 1, wherev X and(2.1) holds,andasequenceoflinear functions

{Urn

m 2 (called partialsum operatorforthe basis

{i,

>

})

can be definedby

Urn(v)-

Z

fi

(v)

i,

i=l The residual operators

{Qrn,

m

>

aredefinedby

vX,m_>

1.

Qm(v)

v

Um(V),

v

,

m

>

1.

A

Schauder basis is saidtobe a

monotone basis

if

IUm(v)

I,

m

>

isamonotone sequencefor each v X.

A

sequenceof random elements

{V

n,n

> 1}

in a normed linear

space X

is saidtobe

(uni-formly) tight if for eachE

>

0,there exists a compact subset

K

eof

X

such that

P{V

Kt}

2 E foralln

>

1.

LEMMA

(AdlerandRosalsky [1]). Let

X

_oand

X

be random variablessuch that

X

_{o is} sto-chastically dominated by

X

inthesense that there exists aconstant

D <

such that

Then for allp

>

0

P{

IXol

> t} <

DP{

DX >

t}, t>O. (2.2)

EIXolrrI(IXol

<

t)

< DtPP IDXI

> t}

+ Dp+IEIXIPI(IDXI

<

t),

t>O.

(2.3)

LEMMA

2 (Adler and Rosalsky [1]).

Let

IX

n

> 1}

and

X

be random variables such that

{X

n

>

1}

isstochasticallydominatedby

X

in the sense that there exists aconstant

D <

such that

P{

IXnl

>

t}

DP{ IDXI >

t},

>

0,n 1.

Let

{ca,

n

1}

bepositiveconstantssuchthat

[nax

cff]

---1

O(n)forsomep

>

0and

J _jn

cff

P{IXI >

Den}

<oo. Thenforall0< M <-0,

(6)

512 Ao ADLER, A. ROSALSKY AND R.L. TAYLOR

----"

EIXnIP

I(IXnl <

Mc

a)

<

n=l

Cn

p

LEMMA

3.

Let

X

_{o and}

X

be random variables suchthat

X

_oisstochastically dominatedby

X

inthe sense that(2.2) holds. Then

and

EIXolI(IXol

>x)

f

P{

IXol

>t}dt

+

xP{

IXol

>x}, x

>

0 (2.4)

EIXolI(IXol

>

x) _<

D2EIXII(IDXI

>

x), x

>

0.

(2.5)

PROOF. Integration byparts yields(2.4), andthen(2.5) followsimmediately from (2.4)and

(2.2).Vl

LEMMA

4(AdlerandRosalsky

[2]). Let X

be a random variable such that

P{ IXI

> t} is reg-ularly varyingwith exponent

p

<-1. Then

X

Lp

for all0

<

p <

-p

and

EIXII(IXI

>t)=(l+o(1))

P+I

tPIlXI

>t}

ast--->,,,,.

Thenextlemma shows that stochastic dominance canbe accomplished byasequenceof ran-dom variableshavingabounded absolutep-thmoment(p> 1).

LEMMA

5 (Taylor [3, p. 123]). Let {X n_> _1} _{be random variables such that}

su

n>_

El

Xnl

P

<

_{for some}_p

>

_{1. Then there exists a random variable}

X

_with

E

IxIq

<

for all 0 < q < psuch that

PI IXnl

>

t}

<

P{

IXl > t},

>

0, n > 1.

Finally,aremark about notation is in order.Throughout,the symbol

C

denotes a generic con-stant(0

< C <

*,,)which isnotnecessarilythe same one in eachappearance.

3.

SLLN’S

UNDER

PROBABILISTIC

AND

GEOMETRIC CONDITIONS.

Withthesepreliminaries accounted for,the firstgroupofresultsmaybe established. The ran-dom elementsareassumedtobeindependent, andgeometric conditions are

placed

on the real

sepa-rablenormed linearspace. The

space

isassumedtobe a BanachspaceofRademachertypep (for

suitablep)in Theorems 1-7,anditisassumedtobe Beck-convex ha Theorem 8. Thenextlemmais thekey lemma inestablishingTheorems1-4.

LEMMA

6.

Let {V

n,n

> 1}

be independent random elementsin a real

separable,

Rademacher

(7)

ran-dom elementVinthe sense that(1.2)holds. Let

{a

n and

{bn,

n:> beconstants satisfy-ing0

<

b

T

and

(3.1)

x-"z_,p{

lanV]

>

Dbn}

<

0% (3.2)

then

Zaj(Vj- EVjI(I lajVjl

<

D2bj))

j=l

--

0a.c.

(3.3)

PROOF. Let

Cn-anl

Yn

VnI(llVnll

<

D2cn),

n _> 1. (3.4) Thenfor n

>

aj(Yj- EYj)

[I p

(since

X

is ofRademachertypep)

o(1) (by

Lcmma

2),

whence

EII

-SI

IP--

0

=I

bj

for some random clementSin

X

implying

n

X

aj(Yj

EYj)

p_.

S.

j=

bj

Sinceconvergence inprobability and a.c.convergencearccquivalcntfor sums ofindcpcndcnt ran-domelements in aseparableBanach

space

(seeIt6 and Nisio[14]),

.,

aj(Yj-

EYj)

implyingvia theKronecker lemrna that

converges a.c.

X

EV)

(8)

514 A. ADLER, A. ROSALSKY AND R.L. TAYLOR

However,

P{liminf

IV

Yn]}

bythe Borel-Cantelli lemma since(1.2) and(3.2)ensure that

P{Vn* Yn}

P{llVnll

>

D2cn}

< D

,

P{IIVIi

>

Dcn}

<

,.

n=l n=l n=l

The conclusion(3.3)then followsdirectlyfrom(3.5).

D

In

the firsttheorem,there is atrade-offbetween theRa(lemachertype and the condition(3.6);

thelargertheRademachertypep,the condition(3.6)becomes less stringent (since

bn/la

’).

THEOREM 1. Let

{Vn,

n2 beindependent random elementsin arealseparable,

Rademacher typep (1 _<p < 2)Banach

space.

Suppose

that

{Vn,

n

>

1} isstochastically dominated

by arandomelementVinthe sense that(1.2)holds. Let

{an,

n

>

and

{b

n bcconstants satisfying 0

<

b

T

,,,,,

bn/la

T,

b

-lajlP

-I

an

I’P

jn’-jP

O(n)’ (3.6)

and

lanl

j__l’j

O(n)"

If theseriesof(3.2) converges,then theSLLN

(3.7)

obtains.

Z

aj(Vj-

EVj)

-->0a.c.

PROOF.

Define {c n

>

1} and

{Yn,

n

>

1} asin(3.4). Note attheoutsetthat(3.7)ensures that c

< Cn,

n

>

1, and so for all

>

1,by (1.2)and

(3.2)

,

PII

IVjl

>

CD2n}

< D

,

PII

IVI

>

CDn}

n=l n=l

< D

P{I

IVl

>

DCn}

<

,,,,.

n--’l

Thus, Ell

Vjll

<

,,

>_ 1,and so (see, e.g., Taylor [3, p.

40])

Vj,

>

1, all haveexpectedvalues.

Also, c

T

by(3.6).

Next,

(3.3)holdsby Lemma6 and soitonly needstobedemonstrated that

ajEVjI(I]Vjl]

>

D2cj)

(9)

Tothisend,

El

IVnl

II(I

IVnll

>

D2cn)

n=l

Cn

<_D2

_1

EIIVIII(ilVII >Dc

n)

(by (2.5))

n=l Cn

D2

.1

El

IVI

II(Dcj

<

IVII

<

Dcj+

1)

n=l

Cn

j---n

j+l

<

D

2 _El

IVI

II(Dcj

_<

IVII

<

Dcj+l)

n=l

Cn

<

D3

cj+l

PIDc

< lVll

<

Dcj+)

(by

(3.7))

J=- cj+

< C

jPIDcj

<

lVll

<

Dcj+}

j=l

C

PIDcj

<

lVll

<

DCj+l}

j=l n=l

C

E

PIDcj

<

lVll

<

Dcj+l}

n=l

C

P{IIVII >

Den}

<** (by(3.2)),

n=l

whencebytheKronecker lemma

I

aj

EVjI(I

IVjll

>

D2cj)l

lajlEI

IVjl

II(I

IVjll

>

D2cj)

j-I j-I

o(1).l-I

REMARK.

Apropos

of Theorem1,the authors are abletoshowthrougha slight modification

ofthe argument thatthecondition

bn/lanl

T

can be

replaced

bythesomewhat weakercondition

bn/la

n O(inf

bj/l

aj

I).

THEOREM

2.

Let

{Vn,

n

>

be independent random elementsin a real

separable,

Rademacher type p

(1

<

p <

2) Banach

space.

Suppose

that

{Vn,

n

> I}

isstochasticallydominated

byarandom element

V

inthe sense that

(1.2)

holds,and

suppose

thatElIVll

<

.,,.

Let

la

n,n

>

and

Ib

n,n

>

beconstantssatisfying 0

<

bn

’1"

**,(3.1),and

ajl

O(bn).

j=l

If theseriesof

(3.2)

converges,then the

SLLN

(3.8)

obtains.

(10)

516 A. ADLER, A. ROSALSKY AND R.L. TAYLOR

PROOF.

Define

{c

n,n

>

1} and

{Yn,

n

> 1}

as in

(3.4). Note

attheoutsetthat(1.2)

guaran-teesthatEll

Vnll

<

,

n 1, andso

V

n,n 1,all have

expected

values.

Now (3.3)

holdsby

Lemma

6and so itonlyneedstobedemonstrated that

aEVI(

V

>

D2c)

j=l

--0.

bn

To

thisend,notethat

(3.1)

ensures c

--

-0,whence by

(2.5),

ElIVll

<

**, andthe Lebesgue dom-inated

convergence

theorem

II

EVaI(I

V

II >

D2Cn)l

Ell

V

II

I(ll V >

D2Cn)

g

D2EI

IVI II(I

IVll

>

Dc

n)

o(1).

But

thenby

(3.8)

andtheToeplitzlemma

II

ajEVjI(IIVjll

>

D2c)II

lajl

lEVjI(IIVjll

>D2cj)II

b_n

bn

o().

r

THEOREM3. Let

{V

n beindependent randomelements in areal

separable,

Rademacher typep (1

<

p

< 2)

Banach

slSacc.

Suppose

that

{V

n,n

>

is stochastically dominated

byarandomelement Vinthe sense that(1.2) holds, and

suppose

that

P{I

IVl

>

t}

isregularly varyingwith exponent

p <

-1. (3.9)

Let {a

n

>

and

{b

n,n

>

1 beconstantssatisfying 0

<

b

’1"

and(3.1). Iftheseries of(3.2)

converges,

then the

SLLN

obtains.

b

--)0a.c.

PROOF. Define

{c

n,n

> 1}

and

{Yn,

n

> 1}

asin

(3.4).

Now El

IVII

< by Lemma4 and

so

(1.2)

ensuresthat Ell

Val

<

00,n

>

1, implyingthat

n,

n

>

1,all haveexpectedvalues. Again

(3.3)holdsby Lemma 6and so itonlyneedstobe demonstrated that

ajEVjI([ Vjl

>

D2cj)

J=l _--)_0.

ba

To

thisend, itfollows from(2.5), (3.1),andLemma4 that for all n

>

some no

El

IVnl

II(I

IVnll

>

D2cn)

<

D2EI

IVI

II(I IVll

>

Dc

n)

<

CcnP{

IVI

>

Den}.

(11)

and so

----1

_El

IVnl

_II(I

IVnll

_>

D2cn)

g

C

+

C

PIIIVII

> Dc

n)

<

n=l Cn n

I1

ajEVjI(I IVjll

>

D2cj)ll

lajlEI IVjl

II(I

IVjll

>

D2cj)

j=l

S j=t o(1)

bn

by the Kronecker lemma.I"1

REMARK.

Apropos

ofTheorems 1,2,or3,

Example

of Adler and

Rosalsky [2]

shows that the Theorems canfail withoutthe

assumption

(3.7), (3.8),or(3.9), respectively.

Theensuinglemma can be

helpful

inverifying the conditions

(3.6), (3.1), (4.6)

ofTheorems 1, 2, 3,or 11, andit willbe usedintheproofof Theorem 4.

LEMMA

7

(Adler

and

Rosalsky

[1]).

Let

{Cn,

n

> 1}

beconstantswith0

<

cnP/n

"1"

for some

p

>

0. Then

iff

lim inf

cpm

>

r for someintegerr

>

2.

tl---

cP

Thenexttheorem is a random element version ofaclassicalresultof Feller[11 whichhad extended theMarcinkiewicz-Zygrnund SLLNtomoregeneral normingconstants.

THEOREM

4. Let

{Va,

n bei.i.d,random elements inarealseparable,Rademacher type

p (1 < p

<

2)Banachspaceandlet {bn,n

> 1}

bepositiveconstants.

Suppose

that either

bn

(i)

EVI=0,

,1,,

---’1’

for

somex>--n _na _p

or

(ii)

E

V)

**,

"I’.

n If

P{IIVIII

>b

n}

<*,,, (3.10)

then

j=l

(12)

518 A. ADLER, A. ROSALSKY AND R.L. TAYLOR

PROOF. In

either case b

"r

and

blain

’.

Now

bn/nI

"

where otin case (i) and

I

in case(ii). Thus,

and soby Lemma 7

b’n

(2n)lP

₂Ip _2,

liminf--- >liminf

>

n-oo-

bnP

n--

nlP

Thenbyl..emma6

bnP

Z

O(n).

(Vj

EVjI(I IVjll

_<

hi))

j=l

Oa.c.

bn

In

case(i),

bn/n

,l,

and

(3.10)

entail(scc ChowandTichcr [10,

pp.

123-124])

(3.12)

j=l

<

j=-I _o(I)

b

bn

which when

combined

with

(3.12)

yields

(3.11)

since

EV

0.

In

case(ii), in view of

(3.10),

necessarily

bn/n

’l"

and so(seeChow and Teicher

[10, pp.

123-1241)

II

EVjI(I

IVjll

<

bj)ll

j=l

b.

yielding (3.11)via(3.12).v!

o(1)

REMARK. In

the special case where

EV

0,

EI

VI

q_< _{for some}

<

_q

<

_p

<

_2,_and b n

l/q,

n

>

1,Theorem4(i)reducestothe Marcinkiewicz-Zygmund type

SLLN

Vj/nI/q 0a.c.of Woycz’yfiski

[15]. Woyczyfiski’s

resulthasbeen improved

by

de

Acosta [16].

Forsomerelated results, see

Wang

andBhaskaraRao 17].

THEOREM

5.

Let {V

n,n

> 1}

beindependentrandom elements in a real separable,

Rademacher typep (1

<

p < 2)Banachspaceandsupposethat

su"

El

IVnl

IP

Let {a n_> 1} and

{bn,

n_> be constantssuch that0<b

’I"

and

Then theSLLN

an

O(n-I/P(log

n)

-I/q)

for some 0< q< p.

bn

(3.13)

(13)

obtains.

aj(Vj- EVj)

j=l

brl

---) 0a.c. (3.15)

PROOF.

Condition

(3.13)

ensures thatVn,n

>

1, all haveexpectedvalues. Letc

bn/lanl,

Yn

VnI(I

IVnll

< Cn),

n> 1. Now by (3.13)and(3.14)

El

IYnl

p _El

IVnl

IP

1

Z

-<Z

_<cz

<

n=

c.r’

.=,

c

.=,

c

implying (see theproofofLemma 6)

(3.16)

Now

aj(Yj- EYj)

j=l

b ---) 0 a.c.

(3.17)

El

IVnl

IP

)",P{V,

#

Y,}

PIIIV,

>

%1

<

(3.18)

n=l n=l n=l CnP

recalling (3.16), whence by theBorel-Cantellilemma P{liminf

[V

n

Yn]}

implyingvia

(3.17)

that

aj(Vj- EYj)

j=l

0a.c. (3.19)

Next,

Z

__1

EliVnllI(lIV,all >ca)

n=l

Cn

Y’P{

’Vnl’

>

Cn} +

X

"n]

P{I

’Vnl’

>

t}dt

n--I

Cn

IP

(by

(3.18))

<c+cz

1

n=l

CnP

(by

(3.13)

and(3.16)),

and sobythe Kronecker lemma

(by

(2.4))

1

ajEVjI(lIV1ll

>

cj)

ll

b

yielding(3.15)via

(3.19).

E!

lalEI

IVjl

II(I

IVjll

>

cj)

b o(1)

THEOREM6.

Let {V

n

>

be independent random elements inarealseparable,

(14)

520 A. ADLER, A. ROSALSKY AND R.L. TAYLOR

_<q < p. Let

{a

n

>

and {b n >_ beconstantssatisfying0<b

"

,,

(3.8),and ThentheS

LLN

an

O(n-l/q).

bn

obtains,

ba

(3.20)

--) 0a.c.

(3.21)

PROOF.

Note

that(1.2)entails

E

lIVnJlq

<0% n

>

1, and henceV n

>

1, allhaveexpected values. Letc

bn/I

a

I,

Yn

VnI(I

IV

<

nl/q),

n 1.

Now

El

IYnl

IP

np/q

nl/q

<D

P{IIDVII

>

.=

c

.=

c

4-

Dp+

_1

El

IVI lPI(I

IDVll n

/q)

(by (2.3))

-1=

C/

C

n’-q

El

IVI IPI((k-1)

uq

<

lDVll k

uq)

n=l 1-1

(by (3.20)andEll V Iq<,,o)

C

+

C)’

El IVl

IPI((k-1) vq <

IDVII <

k

l/q)

n"-q

k--1 n----k

<

C

+

C k(q-p)/qElIVI

IPI((k-l)

l/q_<

IDVI

<

_k

/q)

k-=-I

<

C

+

C

EI

IVI

IqI((k-l)

uq < IDVII <

k

uq)

k=l

C

+

CEI IVI

Iq

<

implying (see the

proof

of

Lemma 6)

ajCgj- EYj)

b_n

--

0 a.c.

Now

by (1.2)andElIVl q

<

0%

PlVn

Yn}

P{

IVnl

>

nl/q}

<

D

P{

IDVI

>

n

l/q}

<

*%

n=l n=l n=l

and sobythe Borel-Cantelli lemmaP{liminf

[V

Yn]}

implyingvia

(3.22)

that

(3.22)

aj(Vj-

EYj)

j=-l b

---) 0a.c.

Next, by (2.5),

EIIVII <

,,,

and theLebesguedominatedconvergence theorem El

lVnl

II(I

IVnll

>n

I/q) <

D2EI

IVlII(I

IDVII

> n

I/q)

o(I),

whenceby

(3.8)

andtheToeplitzlernrna

(15)

bn

yielding

(3.21)

via(3.23).

laIEl

IVjl

II(l

IVjll

>

jl/q)

<

j=l o(I)

Thefollowing

Corollary

isan extensionofTheorem2of Adler and

Rosalsky [2]

(which,in tum, is an extension ofTheorem3.1 ofFernholz and Teicher[18])and establishesaSLLN for normed weighted sums ofstochasticallydominated random variables. Itwill be used in the

proofs

of Theorems8 and9butmaybe ofindependent interest.

COROLLARY

1. Let

{X

n

>

1}

beindependentrandom variables and let

X

be an

L

_v ran-dom variableforsome

<

p

<

2.

Suppose

that

IX

n

>

1}

is stochastically dominatedby

X

inthe sense that there exists aconstant

D

< suchthat

P{

IXnl

>

t} < DP{ IDXI

> t},

>

0,n

>

1.

Let

{a

n

>

and

{bn,

n

>

beconstantssatisfying0

<

b

T

*,,,

an/b

O(n-VP),

and(3.8). Then theSLLN

obtains.

b --)0 a.c.

PROOF. Since(R, I’!)isa realseparable,Rademacher type 2 Banachspace,theCorollary

fol-lowsimmediatelyfrom Theorem6withp 2andq

p <

2.!"1

THEOREM 7.

Let {V

n

> 1}

beindependentrandom elements in a real

separable,

Rademacher typep (1 <p < 2)Banachspace.

Suppose

that

IV

n> 1} isstochastically dominated

byarandom element

V

inthe sense that(1.2)holds, andsupposethat

E IVI IP

<**. Let

la

n

>

and

Ibn,

n

>

<

b

"1"

,o,(3.8),and(3.14). Then theSLLN

obtains.

0a.c.

PROOF. Using the truncation

Yn

VnI(I

IVn

S

nl/r’),

n

>

1, theargumentisaslight modification of that usedtoestablishTheorem6. The details arelefttothe reader. 121

REMARK. An

interestingquestionwhich weareunabletoresolveiswhether Theorem 7

(16)

522 A. ADLER, A. ROSALSKY AND R.L. TAYLOR

7 should becomparedwithTheorem I0 wherein the

{V

n

> I}

are(uniformly)tight.

ThenextTheorem establishes aSLLNfornormed weightedsumsof random elements in a real

separablenormed linearspacewhich isBeck-convex.

It

shouldbe

compared

withTheorem5 of TaylorandPadgett

[19]

(orTheorem5.3.1 ofTaylor [3, p.

137]).

THEOREM

8. Let

{V

n,n

> 1}

beindependentrandomelementsin a realseparablenormed

linearspacewhich is Beck-convex and let {a n

>

1} and

lb,,

n

>

1} beconstantssatisfying a

n>0,n>l,0<b

nT**,

aj=O(bn),an/b

n=

O(n

-l/p)

forsome

<p<2,

and

j=l

_(aj

d

n)

o(b

n)

(3.24)

whered

n--

min _aj,

n>

If

E IVnllq

l<j

,

< for someq > p,then theSLLN

obtains.

b

--->

0 a.c. (3.25)

PROOF.

Without lossof generality,itmayandwillbesupposedthatEV 0,n

>

1.

Sup-pose,initially, that the

{V

n,n

>

areuniformly bounded in the normby aconstant, that is,

lV.[[

< C

a.c. Then, since

nda

<

a

O(b.),

j-I jl

@.

b.

cz

%-%)

+

0a.c.

b_n n

by

(3.24)

d a

SLLN

ofBeck [20,

eorem

10]

(which is

eorem

4.3.1 ofTaylor

[3,

p.

87])

thereby

pvg

the

eorem

when

(

IV

1

C

a.c.

Next,

in general, define

X.=VnI(IIV.II

<M),

Yn=V.I(IIV.II

>M), n21,

where

< M <

is aconstant.

By

theportionof the theorem

already proved,

Note

thatforn 1,

aj(X-

EXj)

(17)

E{Mq-IIVnlII(IIVnil

>M)}

El

IYnll

Mq-I

El

IVnl

IqI(I

IVnll

>

M)

C

<

Mq_ Mq_

and so in view of aj O(b

n)

j=l

<

j=l

+

j=-I

b

bn

j=l

+

bn

X

aj(I IYjll

El

IYjl

I)

<

j=l

+

bn

2

Z

ajEI IYjll

b

C

Mq-I

Now

{I

Ynll

E

lIYn

II,

n

> I}

areindependentmean0 random variableswith su El

IYnl

lq

<

2q El

IVnl

lq

,l

E

IIYnll

EllYnll

q<2q

n>l

na

By

Lemma 5,thereexists arandom variable

Y

withElY p

<

_{such that}

PtlIIYn’I

-EllYnll

l>t}

<P{IYI >t},

t>O,n>

1, whenceby Corollary

b --> 0a.c.

But

thenby

(3.26)

and

(3.27)

<

limsup

n- b

I,

aj(Yj- EYj),I

j=l

+

lknsup

n-

b.

C

S a.c.

Mq-I

and since

M

isarbitrary, the conclusion

(3.25)

follows.[] 4.

SLLN’S UNDER

PROBABILISTIC

CONDITIONS.

In

this section,

SLLN’s

are obtainedwithoutimposinggeometric conditions on the Banach

(18)

524 A. ADLER, A. ROSALSKY AND R.L. TAYLOR

Banachspaceareassumedtobe (uniformly) tight.

ForaBanachspaceadmittingaSchauder basis, recall the definitions of

If

i,

>

],

U

m,m

>

1},

and

Qm,

m

>

1} presented

in Section 2. Theorem 9 should be comparedwith

Theorem5.1.4ofTaylor [3,

p.114].

THEOREM

9.

Let {V

n,n

>

beindependent,mean zero random elements in a realseparable

Banachspace admittingaSchauder basis

{13

i, > 1}.

Let

{a n

>

1} and bn,n

> 1}

beconstants

satisfying 0

<

b

"

*,,,(3.8),and

an

O(n

-I/p)

(4.1)

bn

for some

<

p

<

2.

Suppose

that there exist randomvariables

{Xi,

and

{Ym,

m2 and a constantD

<

such that

Pllfi(Vn)

> t}

DPIIDXil

> t},

>

0,nR 1,

>

1,

P{

111Qm(Vn)l

El

IQm(Vn>l

> t} <

DP{

IDYml

>

t},

su El P

t

X

il

_<*%

supEIYmlP<,%

m’l and

t>0,

m

l,n> 1,

Then the

SLLN

lim

.

El

IQm(Vn)l

O.

(4.2)

-

0 a.c.

obtains.

PROOF.

It

follows directly

from

Corollary

that

bll

--)0 a.c. for each and

lal(I

IQm(Vj)I

El

IQm(Vj)I

I)

Tin,n_=

j--1 0 a.c. for eachm

>

1.

Then

i-I

b

j--I

b

1113ill

0 a.c. for eachm> 1.

(4.3)

(19)

Thus,by (4.4), (4.3), (3.8),and(4.2)

J=Ibn

<

lUm

t’-bn

II+

II

EajQm(Vj)

j=l b

[

b"

II

+

T=

+

C

El

IQ=(Vj)I

---,

0a.c. as firstn-***and thenm-***,v!

THEOREM

10.

Let {V

n,n

> 1}

be a(uniformly) tight

sequence

of independent,meanzero

randomelementsin areal

separable

Etanach

space X. Let {a

n and

{b

n

>

beconstants satisfying 0

<

bn

’l"

**,(3.8),and

(4.1)

for some 1

<

p

<

2.

Suppose

that

IV

n,n:>

1}

is

stochasti-callydominatedbyarandomelement

V

inthe sense that

(1.2)

holds,and

suppose

that

E

IV

liP

<

**. Then theSLLN

obtains.

0 a.c.

PROOF. Let

hbe anorm-preserving, bicontinuous,linearmapping of

X

into

C[0,1]

(--the Banachspaceof all continuousreal-valuedfunctionsyon

[0,1]

withnorm

II

y ll

ul

ly(t) l).

TheBanach

space

C[0,1]admits amonotonebasiswhere

IQm(y)l

lyll

and

Ifm(y)l <

lyll

for each

y

[0,1]

andm

>

andwhere

IQm(y)l

I,

m 1 is amonotonedecreasing

sequence

for eachy C

[0,1].

Then

{h(Vn),

n

11

isa(uniformly) tightsequenceofindependent,mean zero randomelementsin

C[0,1]. Now

for arbitraryI

>

0, chooseu

>

0 sothat

D2EII

V

llI(11Vll

>

u)

<

-.

ThenLemma 3 provides Ell

Vnl

II(I

IV,

II

>

u)

<

-

for all n 1.

By

(urfiform) tightness,acompactsbset

K

of

C[0,1]

ma,y

bechosensothat

PIh(V,)

d

KI

<

3"-’"

for all n 1,whenceEl

IVnl

II(I

IVnll

_<_u)I(h(V

n)

a

K)S for all n

>

1. Since

IQm(y)l

foreachyinthe compactset

K,

Dini’stheoremensures thatthere exists anintegerm_osuchthat

sup

Q=(y)

<

e

yK frall m

>

rn"

Thenfrall m

>

rn

and n

>

EI

Q=(h(Vn))l

< EI

Qm(h(Vn))I(I

IV

S

u)I(h(Vn)

K)

(20)

526 A. ADLER, A. ROSALSKY AND R.L. TAYLOR

therebyestablishing(4.2) for the randomelements

Ih(Vn),

n

>

}.

Theidentifications

X

IVl and

Ym

IVI

+

DE

IIVll for all

>

and m

>

ensure that the other conditionsofTheorem9 hold. Thus

ZajVj

h(ZajVj)

Zajh(Vj)

j-I j=l j--I

bn

--->0 a.c.V1

REMARKS.

(i)When a_n 1,b_n n, n

>

1,and Ell

V

II

P

<

_{for some}_p

>

_{1, Theorem} 10inconjunctionwith

Lemma

5will establishtheSLLN ofTaylor [3, Corollary 5.2.9, p.

133]. As

pointedoutby Taylor [3,

p. 133],

that same

SLLN

canbe obtained from Theorem5.2.8ofTaylor [3, p. 13I]butunder the stronger assumption that

s

Ell

Val

P

<

_{for some}_p

>

_2.

(ii)Theorem 10canfail ifp andEl IVll

,,,,.

Foranexample,seeTaylor [3, Example5.2.3,

p. 135].

ThenextCorollary should becomparedwithTheorem5.2.8 ofTaylor

[3,

p. 131 ].

COROLLARY

2.

Let {V

n,n >_

I}

bea(uniformly) tightsequenceofindependent,mean zero

randomelements in a realseparableBanachspace.

Let

{an,

n

> 1}

and

{b

n

>

1}

be constants satisfying0

<

b

’1"

**,(3.8),and(4.1)for some

<

p

<

2. If

thenthe

SLLN

Ell

Vnll

q

<

forsome

q

> p,

(4.5)

obtains.

ZaV

b 0 a.c.

PROOF.

Condition(4.5)ensuresby

Lemma

5that(1.2)obtainsandEl IVl p

<

.

The Corollarythenfollows from Theorem 10. 121

In

thenextCorollary,thesequence {Va,n :>

1}

is i.i.d, and themomentcondition(4.5)is

weakenedtoEll

VIIIP.

TheCorollaryshould becomparedwithTheorem5.1.3ofTaylor [3,

p. 112].

COROLLARY 3. Let

{V

n

>

be i.i.d,_{mean zero random elements in}arealseparable

Banachspace. Let

{an,

n

> 1}

and {bn,n

> I}

<

b

"1"

*,,,(3.8), and(4.1)

(21)

obtains.

bn

---) 0a.c.

PROOF. Since thei.i.d,hypothesisensures that

{V

n

> 1}

is automatically (uniformly)tight

(see Taylor[3, p. 121]),theCorollary follows immediately from Theorem 10.[]

REMARKS.

(i)

In

theparticularcase where a 1, b

---

n, andp 1, Corollary 3reducesto theSLLNofMourier[7].

(ii)

A

Fr6chetspace_is acompletelinear metricspace. Using Theorem 10,a

SLLN

maybeobtained for random elements in arealseparableFr6chetspace

F

which is alocallyconvexspacewitha countable family of seminorms

{Pk,

k

>

1} defined on itsuchthatthemetricdis definedby

d(x,y)

Pk(x

Y)

k=l

2k(1

+

pk(X y))

forx,y F.

Thedetails will notbe givensincethe argumentparallels thatof Theorem5.2.10ofTaylor [3, p. 136]. (Corollary2plays the same role in theproofasTheorem5.2.8 ofTaylor [3,p. 131] played in

provingTheorem5.2.10.) Infact, almost all of the resultsirt this sectionhaveparallelresults for Fr6chetspaces.

Inthelast theorem, there is noindependence assumption onthe sequenceof random elements.

Moreover,

the spaceis equippedwithaseminorm

p

whichis notnecessarily a norm and thus the resultisapplicabletoalargerclass ofspacesthanrealseparable normedlinearspaces.

"Eae

definitionofrandom element is

analogous

tothat discussed in Section forreal

separable

normed

linearspaces.

THEOREM

11. Let

{Vn,

n

> 1}

be random elements in a realseparableseminormedlinear

space

with seminorm

p. Suppose

that

{Vn,

n

> 1}

isstochasticallydominatedbyarandomelement

V

inthe sense thatthere exists aconstant

D <

such that

P{p(V

n)

> t} < DPIp(DV) > t},

>

0,n

>

1.

Let

an,

n

> 1}

and {b n

> 1}

beconstantssuch that0

<

bn

"1"

and

SjSn

lajl

Jj-’gTJ

(22)

528 A. ADLER, A. ROSALSKY AND R.L. TAYLOR

hen

P{p(anV)

>

Db

n}

<

*, n--1

bn

PROOF. Set

Yn

P(Vn),

n _> 1,and

Y

p(V).

Thenby Theorem 2 of AdlerandRosalsky [1],

p

ajVj

ajl p(Vj)

:

j--1

bn

---,

0 a.c.r"l

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