AND RANDOM

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.

1980

AMS SUBJECT CLASSIFICATION CODES.

Primary 60B12, 60F15;

Seconda

^00,.

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 results^furnishconditions 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 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

areestab- lished under

geometric

conditions on Xwhereas in Section4,

SLLN’s

areestablishedwithout such conditions. The

SLLN

problem^wasstudiedbyAdler andRosalsky [1, 2]in the random variable case, andsomeof those results will be extendedtothe random element case^{in the}current work.

Taylor

[3] provided

^a

comprehensive

^andunified treatment of results under whose conditions

anjVj

^{---> 0a.c. where}

^{V

^{n,n >_ are}independent, mean zero random elements in a realsepa- rable normed^linear

space

and

{an

_j,

^{< <}

^{n, n}

^{> 1}}

^{is a}triangular array of^constants. Someo: tbc

argumentsinTaylor’s monograph utilizeda result ofRohatgi

[4]

whichwillnow be stated.

(Rohatgi’swork generalized earlier work of Pruitt

[5].)

THEOREM

(Rohatgi [4]).

Let {Xn,

ⁿ

I]

be

independent,

meanzero 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 where2

>

p

---"

^{_> 1.}

For

example,for^i.i.d,random variables

IX

^butnnot

^>

^{fromwithRohatgi’s}

EX1

^0,theorem (which wouldthe classical

Ko.lmogorov

require

^SLLN EX

^{2< 00).}

Xj/n -

^{0 a.c.follows from}

^Corollary

A

SLLNfor normedweightedsumsof random elements in a real separable Banachspacehas beenproved byMikosch and

Norvai’a

[6],buttheft result and thecurrentonesdo notentaileach other.

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, random}elements in a realseparableBanachspaceand ifEl

IVtl ^<

^*,,,then

(Vj EVI)/n

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

j=l

discoveredby Cuestaand

Matran [9].) For

randomvariables,theKolmogorov

SLLN

wasgeneral- 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 are^i.i.d,random variables with

P{YI P{YI

^{-I 1/2.} Let X bearealseparableBanachspaceand let

X X

x

XxX x

and define

C(X)

{v

n

> 1} X**" Ynvn ^converges

ⁱⁿprobability

n-1

Let <

_p

<

2. Then

X

is saidto beofRademacher_{typ_e}pifthere 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 separableBanachspace^isofRademachertypepiffthere exists a constant0

< C <

such that

El

z_,Vjl ^{It’ <} CT_.,

^El

IVil ^It’

foreveryfinite collection

{Vt

^{V of}independentrandomelementswith

EVj

^0,

El

IVjl ^{It <} ,,,,

<jn.

Ifa realseparableBanachspaceis ofRademacher_type

p

for some

<

p

<

2,then it is of Rademachertypeqfor all

<

_q

<

p.

Every

realseparableBanachspaceisofRademacher_{type (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.

A

normed linear

space

X is saidtobe Beck-convexif there exists aninteger N

>

and a number0<I< such that for all choices of

{v

^v

N}

X ^with

lvjll

^{< 1, < <}

^N,

I1+v

__.

:!:

VNI

< N(1-E)

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

A

real separableBanach spaceisBeck convex iff it is ofRademacher 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

there_{exists a}unique 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

J

cff]

_jn

^---1 cff

^O(n)forsomep

^>

^0and

P{IXI >

Den}

^{<oo. Thenforall0}< M <-0, n=l

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} ^isreg- 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-th^moment(p> 1).

LEMMA

5 (Taylor [3, p. 123]). Let {X ^n_> 1} be random variables such that

su

n>_

El

^Xnl

^P

^<

^{for somep}

^>

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 realsepa- rablenormed linearspace. The

space

isassumedtobe a BanachspaceofRademachertypep (for suitablep)in Theorems 1-7,anditisassumedtobe Beck-convex ha Theorem 8. The^nextlemmais thekey lemma inestablishingTheorems1-4.

LEMMA

6.

Let {V

n,n

> 1}

be independent random elements^{in a real}

separable,

^Rademacher typep (1

<

_p

< 2)

Banach

space

X.

Suppose

that

IV

n,n

>- 1}

isstochasticallydominatedby^{a ran-}

dom elementVinthe sense that(1.2)holds. Let

{a

ⁿ and

{bn,

^{n:> beconstantssatisfy-} 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 ofindcpcndcntran- domelements in aseparableBanach

space

(seeIt6 and Nisio[14]),

., ^{aj(Yj- EYj)}

implyingvia theKronecker lemrna that

converges a.c.

X EV)

b

--

^{0a.c. (3.5)}

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 be}independent 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)

b

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 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 Theorem}1,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 by^arandom element

V

inthe sense that

(1.2)

holds,and

suppose

thatElIVll

< .,,. _Let

la

^n,n

^>

^and

Ib

^n,n

^>

^beconstantssatisfying 0

<

b_n

’1"

**,(3.1),and

ajl O(bn).

j=l

If theseriesof

(3.2)

converges,then the

SLLN

(3.8)

obtains.

b_n ^--)0a.c.

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

^inatedthis

convergence

end,notethattheorem

(3.1)

ensures c

--

^{-0,whence by}

^(2.5),

^ElIVll

^<

^{**, andthe Lebesguedom-}

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 have}expectedvalues. 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}.

Thenby (3.2),

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 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 SLLN^tomoregeneral 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 bn

(i)

EVI=0, ,1,, ---’1’

for somex>--

n n^a p

or

(ii)

E V)

**,

"I’.

n If

P{IIVIII

>b

n}

<*,,, (3.10)

then

j=l

bn ^--*

^{0 a.c. (3.11)}

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

PROOF. In

either case b

"r

and

blain ’.

Now

bn/nI "

^{where ot}in 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 Chowand}Tichcr [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 constants}such that0<b

’I"

and

Then theSLLN

an O(n-I/P(log

n)

-I/q)

for some 0< q< p.

bn

(3.13)

(3.14)

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 Cn^P

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,

Rademacher typep (1

<

p 2)Banachspace.

Suppose

that

{Vn,

ⁿ

>

1} isstochasticallydominated by arandom elementVinthe sense that (1.2)holds, and

s.uppose

^thatEllWll^q< forsome

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 theLebesguedominated}convergence theorem El

lVnl

^II(I

IVnll

^>n

^{I/q) < D2EI}

^IVlII(I

^IDVII

^{> n}

^I/q)

^o(I),

whenceby

(3.8)

^andtheToeplitzlernrna

(3.23)

bn 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

_vran- 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} is}stochastically dominated byarandom element

V

inthe sense that(1.2)holds, andsupposethat

E IVI IP

<^**. Let

la

ⁿ

^>

^and

Ibn,

ⁿ

^>

^beconstantssatisfying 0

<

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 holdswith(3.14) replaced bythe somewhat weaker condition

an/b O(n-UP). Moreover,

Theorem

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

7 should becompared^withTheorem 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. b.

@.

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)

b ^---)0 a.c. (3.26)

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 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}}

^areindependent^mean0 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)

(3.27)

<

_limsup

n- b

I, aj(Yj- EYj),I +

lknsup j=l

n-

b.

S

C

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 space.

As

in Section3,momentconditions areplaced^onindependentrandom elements andrestric- tionsareplacedontheconstants

{an,

ⁿ

>

and

{bn,

n

>

}. InTheorem9,the Banach spaceis assumedtoadmit aSchauder basis andinTheorem 10,theindependentrandomelementsin a

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

t

El

^{X il}

^P<*%

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)

(4.4)

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 then^m-***,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

<

b_n

’l"

**,(3.8),and

(4.1)

for some 1

<

_p

<

2.

Suppose

that

IV

^n,n:>

^1}

^isstochasti-

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)

+ElIV

nllI(llV nIl <u)I(h(V,)aK)+EIIVnllI(llV"ll

^>u)<

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 bn bn

^{--->0 a.c.V1}

REMARKS.

(i)When a_n 1,b_n n, n

>

1,and Ell

V II

^P

<

for somep

>

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 somep}

^>

^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}

beconstantssatisfying 0

<

b

"1"

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

<

p <2.If El

IVll ^IP

^<

,,

then theSLLN

obtains.

bn

^{---) 0a.c.}

PROOF. Since the^i.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 a}completelinear metricspace. Using Theorem 10,a

SLLN

maybeobtained for random elements in arealseparableFr6chetspace

F

which is alocallyconvexspace^witha 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 equipped^withaseminorm

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

<

b_n

"1"

and

SjSn

lajl Jj-’gTJ ^=O(n).

^(4.6)

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 bn

---,

_{0 a.c.}r"l

REFERENCES

1.

ADLER, A.

and

ROSALSKY, A. Some

General

Strong Laws

for Weighted

Sums

of Stochasti-

cally

DominatedRandom Variables,

Stochastic

^Anal.

Appl.

(1987), 1-16.

2.

ADLER, A.

and

ROSALSKY, A. On

the

Strong

Law of

Large

Numbersfor NormedWeighted

Sums

of

I.I.D.

RandomVariables, Stochastic Anal. Appl.

i

(1987), 467-483.

3.

TAYLOR,

R.L.

Stochastic Convergence of

^Weighted

Sums

of

Random Elcmcnt in

Linear

Spaces. Lecture

N0tf$

in

Math. 672, Springer-Verlag,Berlin

1978.

4.

ROHATGI, V.K. Convergence

of Weighted

Sums

of

Independent

Random Variables,^Proc,

CambridgePhilos.

Soc.

69

(1971),

305-307.

5.

PRUITr,

W.E. SummabilityofIndependentRandom Variables,[.Math. Mech. 15(1966), 769-776.

6.

MIKOSCH,

T.and

NORVAISA,

R. OnAlmost

Sure

Behavior of

Sums

ofIndependentBanach

Space

Valued Random Variables, Math. Nachr,

125

(1986), 217-231.

7.

MOURIER,

E. El6ments Al6atoires dans un

Espace

deBanach,

Ann. Inst. Henri

^Pincar6,

^Sect.

B, 13 (1953),

159-244.

8.

LAHA, R.G.

and

ROHATGI, V.K.

Probability Theory_.JohnWiley,

New

York, 1979.

9.

CUESTA, J.A.

and

MATRAN, C. Strong

Lawsof

Large

Numbers in Abstract

Spaces

via Skorohod’s Representation Theorem, Sankhy’,

Ser. A,

48

(1986),

98-103.

10.

CHOW, Y.S.

and

TEICHER, H.

ProbabilityTheory: Independence,Interchangeability.Mar- Springer-Verlag, New York, 1978.

11.

FELLER, W. A

LimitTheoremfor Random Variables with Infinite

Moments, Amer. J.

Math.

68

(1946),

257-262.

12.

HOFFMANN-JGENSEN,

J. and

PISIER, G.

The Lawof

Large

Numbers and the Central LimitTheoreminBanach

Spaces,

Ann. Probab. 4(1976), 587-599.

13.

GIESY,

D.P. OnaConvexity Condition in Normed Linear

Spaces, Trans. Amer.

Math.Soc.

125 (1966), 114-146.

14.

1T,

K. and

NISIO,

M. Onthe

Convergence

ofSumsofIndependentBanach

Space

Valued Random Variables, Osaka

J.

Math.5 (1968), 35-48.

15.

WOYCZYNSKI, W.A. On

Marcinldewicz-Zygmund

Laws

of

Large

Numbersin Banach

Spaces

and RelatedRatesof

Convergence,

Probab. and Math. Statist. (1980), 117-131.

16.

DE ACOSTA,

A. Inequalitiesfor B-Valued Random

Vectors

withApplicationstothe

Strong Law

of

Large

Numbers,

Ann.

Probab.

9

(1981), 157-161.

17.

WANG, X.C.

andBHASKARA

RAO,

M. SomeResults on the

Convergence

of Weighted Sumsof Random Elements inSeparableBanach

Spaces,

Studia Math.

86.

(1987), 131-153.

18.

FERNHOLZ,

L.T.andTEICHER, H. Stabilityof Random Variables and IteratedLogarithm

Laws

forMartingalesandQuadratic

Forms, Ann. Probab.

⁸ (1980), 765-774.

19.

TAYLOR,

R.L. and

PADGE’VF,

W.J. Stochastic

Convergence

ofWeighted

Sums

in Normed Linear

Spaces,

J. Multivariate Anal. 5(1975), 434-450.

20.

BECK,

A. Onthe

Strong

Law of

Large

Numbers,

In

..ErgodicTheory; ProceedingsofanInter- nationalSymposiumHeldatTulane University,

New

Orleans,Louisiana,October,

1961

(Ed.

F.B. Wright),21-53, Academic

Press, New

York, 1963.