Convergence in mean of weighted sums of {ank} compactly uniformly integrable random elements in Banach spaces

Internat. J. Math. & Math. Sci.

VOL. 20 NO. 3 (1997) 443-450 443

CONVERGENCE IN

MEAN

OF WEIGHTED

SUMS

OF

{a,,}-COMPACTLY UNIFORMLY INTEGRABLE

RANDOM

ELEMENTS IN BANACH SPACES

M.

ORDOIEZ

CABRERA

lepartment

of Mathematical Analysis

University of Sevilla. Sevilla. Spain.

(Received October 23, 1995 and in revised form March 14, 1996)

ABSTRACT.

The convergence in meaaa ofa veighted sum

k

a.k(Xk

EXk)

of random elementsinaseparableBanach spaceisstudied undera newhypothesis which relates the random elements with their respectiveweights in thesum: the

{a..

}-compactly

uniform integrability

of

{X. }.

This condition, which is implied by the tightness of

{X,,}

and the

{a,,k }-uniform

integrabilityof

{[IX,,

II},

isweakerthan thecompactlymiformintegrabilityof

{X,,}

and leads to aresult of convergence inmeanwhichisstrictlystrongerthan arecent result of

Wang,

Rao

and Deli.

KEYWORDS AND PHRASES:

Weighted sums, random elements in separable Banach

spaces,compactlyuniformintegrability,

{,,,

}-compactly

uniformintegra.bility, tightness,

{an,

}-uniformintegrability, convergence inmean.

1991 AMS SUBJECT CLASSIFICATION CODES: 60B12, 60F05.

1.

INTRODUCTION.

Let

(f,

A,

7’-)

beaprobability space, and let

{X,, },

n N[,beasequence of random elements

in a. separable Ba.nach space

(X,

[[.[[),

i.e., a sequence offlinctions from into Xwhich e A-metwa.blewithrespectt()the Borel subsets of

X.

Let

{a,,},k,n e

N, beanarray of real numberswithsup

]a,[ <

.

In

this paper, we deal with the convergence in mean of the sequence of weighted sums

a.(X.

EX

).

S,,

Habitually, thisproblenaof weak convergence, mwellasthe problem ofstrong convergence,

ha beenstudied byconsideringepa,rately thec()nditi(ms onthe sequence

{X.

(relying

more andmorethe initialhyl)ot,h(.sis_{of independence mad identical}

distribution)

and the conditions onthe array

{a,,.

).

Ord6fiez

([1])

obtainsesults,,f_convergenceinmeanandconvergenceinr-mean

(r

(0,

1))

for weighted sulnsofrandom variables(randomelements in

)

by requiringacondition whi rela.tes the random variables

.k’

totheir respective weightsa,,: the

{a,, }-uniform

integrability

Ceshro uniformintegra.bility (see

[2])

as a particularcase.

A

counterexampleshows that the result of convergence inme,’m _does nothold,a,sta.ted,for weighted sumsof randomelements in aseparable Bana.ch space.

In

this paper,weobtainsucharesult,underanewconditionontherandom elements

Xk

concerning their respective weights in S,,:

{X,,}

is required tobe

{a,k

}-compactly

uniformly

integrable. This requerimentisweaker than the requirement ofcompactlyuniformintegrability

usedby Hoffmann-.]orgensenand Pisier

([3])

and Dafter andTaylor

([4]),

and characterizedby

Wang

and

Rao

([5]);

consequently, ourresult extends the result obtainedby

Wang

et al.

([6],

Theorem

3.2).

2.

PRELIMINARIES.

In

the proof of the mainresult,theembeddingofa.separableB.’mach space

X

inaBanach

space withaSchauderbasisxvillbe usedasinTaylor

([7]).

A

sequence

b. },.

fi

NI,

in

X

is a Schauder basis for

X

if for eachx

X

there existsa unique sequence of rea.l numl)ers

{t. }

such thatz

Z

t.b..

WhenaBanach spa.cc

X

hasaSclmuderbasis

b. },

asequence of continuous linear function-als

{.f,,

on

X

canbe definedby

f,,(x)

t,,,n

IN;

thesearecalled the coordinate functionals for the basis b,,

}.

The paxtiM sum operator

U,,

on

X

is defined by

U,,(.’)

Z

fk(x)bt,

and the residual

k--1

operator

Q,,

on

X

by

Q,, (x)

:r

U.

(.:),

forevery x

e

X.

U.

and

Q.

axetwo sequences of

continuouslinear operat,,rson

X

satisfying lim

U,,(x)

xand lim

Q.(x)

0 for everyxfi

X.

A

sequence

{X,,

of rndm elements in a Banach space

X

issaid tobe tight iffor each e

>

0there existsa cnnpact subset

K

of

X

suchthat

sup

P

[X,,

t

K]

<

e.

Let

p

>

0.

A

sequence

{X,, },

nG

N,

of random elementsina,Bh.nach space

X

issaid to be

compactlyuniformlyp-thorlerintegrnbleif for everye

>

0thereexists acompactsubset

K

of

X

such that

,,p

EllX,

ll"Z[X.l,-]

<

:.

(IA

denotes the indicato: of the event

A).

Ifp 1,

{X,,

issaidt,becompactlyuniformlyintcgrable.

For the characteriztion,,fthis concept, werefer t,,

rang

andRao

([5])

and

Cuesta

d

Matrn

([8])"

Let

{.,,

beasequence,,f rand,,melementsin aseparableBach space, dlet

p

>

0. Then,

{X,,

is c,,mpactly uniformly p-th orderintegrable if,and only if,

{X.}

istight

and

{[IX,,I]"}

isuniformly integrable.

CONVERGENCE IN MEAN OF WEIGHTED SUMS 445

3.

COMPACTLY UNIFORMLY

INTEGRABLE RANDOM ELEMENTS

CON-CERNING AN ARRAY.

Ord6fiez

([1])

introduces thefolloving concept:

DEFINITION

3.1.

Let

{a,,},k,n

E

N,

be an m’ra.y of real constants satisfying sup

Z

[a,d <

cx.

A

sequence

{X,}

ofintegrablerandom viables is said to be

{a,}-unifoly

integable (or uniformly int.egrable concerningtheaxray

a

})

if

lim sup

[o,.

[E[X

[I[Ix

I>,,] O. Thefollowingmsertion is easy tocheck:

PROPOSITION

3.1.

Let

{X,,}

beasequence ofuniformlyintegrablerdomvabl. Then,

{X,,}

is

{a,,

}-uniformly

integrablefor allm’ra.ys

{a,,k

sud that sup

[ak [<

.

k

A

sequence of rmdom elements in aseparable Bana.ch space being compactly uniformly

integrableis the natural extension ofa.sequence of random variablesbeing uniformly integrable

(both

definitionsareequiva.lentwhen the Ba.nach spa.ce is finite

dimensional).

To

this

effect,

we

introducethefollowingnotion"

DEFINITION 3.2. Let

{a,,t.},k,n

E

N,

be an re’ray of real constants satisfying

sup

Z

[o.,,[ <

o.

Let

p

>

0.

A

sequence

{X,,},

n

e

N,

of random elements in asepable

Banachspace

X

issaidtobe

{a,,a

}-cmapactly

uniformlyp-th orderintegrableif for

eve

>

0 thereexists a.compact subset

K

of

X

suchthat

k

Ifp 1,

{X,,

issa.idt,,be

{a,,

}-compactly

uniformlyintegrable.

Thefollowing theorem providesasufficient conditionff,rthe

{a,,

}-compactly

uformp-th

orderintegrabilityof X,,

}:

THEOREM

3.1.

Let

{a,,}.k,n

N,

be array of real constts with

sup

[a, <

,

and let p

>

0.

Let

{X,, },

n fi

W,

beasequence of rdom elements ina

se

arable Sanach space

X,

which istight, and such that the sequence

{[[X,,[[v}

is

{an}-unifoly

integrable.

Then,

{X,,

is

{a,,

}-c,,xnpa,’tly

mfiformlyp-thorderintegrable.

PROOF.

By

Theoen 2 in

[1],

given

>

0, thereexists

>

0 such that whenever

{Ak}

is a sequence,.,feventssatisfying sup

]a,,IP(A)

<

6, thensup

[a,,[E[[X[["IA, <

e.

k k

The tightness of

{X,}

implies the existence of a. cmpact subset

K

of

X

such that

P

IX,

I(]

<

C

-

fi,r,,very

,

N,where

C

>

0isany constant such tha.tsup

[a,[

C.

k

Thensup

[",kiP

[X

I(]

<

$, andtherefore:

i.e.,

{X,,

is

{a,,k

}-compactly

uniformly

v-th

order integra.ble.

Itiseasytocheck thefollowing

PROPOSITION

3.2.

Let

{X,

beasequenceof compactlyuniformlyp-thorder

(p

>

0)

integrable randomelements in a.separable Banach space. Then,

{X,,

is as in Theorem 3.1,

andso

{X,

is

{a,,k

}-c,mpactly

uniformlyp-thorderintegrablefor all arrays

{a,,

}

such that sup

E

k

REMARK.

The characterization of

{a,,

}-compactly

unifo.rnp-thorderintegrability of

X,,

interms oftightnessof

{X,,

a.nd a,,

}-uniform

integrabilityof

{ilX,,

’

is not available

(i.e. the condition in Theoren 3.1is not

necessary),

asis shownby consideringthe sequenceof

rmdom elements in

{.’

I"

II.’ll

E

Ix,,I

<

oo}

definedby

X,,

e,,withprobability

1, where e,

}

isthestandard basis of ,’rodthe,’u’ra.y ifl<k<n a,,=

0 ifk>n.

Givene>0,taken

ENsuchtha.t

& <,andlet thecompact

K={e

e2,

,era}

Then

k_>

Therefore,

{X,,}

is

{a,,t.}-compactly

uniformly p-thorderintegra.ble

(p

> 0),

but

{X,,}

is nottight.

4.

CONVERGENCE IN

MEAN.

Ord6fiez

([1])

obtains the h,lloving result of convergence in mean for weighted sums of random variables:

THEOREM

4.1. Let

ta,,},Ic,

N,

bea.narray of real constantssatifying:

k

b)

hmsup

a,

0.

Let

{X, },

n N, bca equence of pairvise independent and

{a,

}-uniformly

integrable

random va.riables.

a,,(Xt.

EX

0 in mean.

Then

S,

This theorem does n(t _hold, as

stated,

for random elements in

separable

Bach spaces

(see

[1]).

Now,

in Theorem 4.2, we prove that such a.nextension is possible fora sequence of raadom elements

{X,,

being

{a,,}-c(,mpactly

unifornfly integra.ble. Previously, weprove the

fl)llowing lemma"

LEMMA

4.1.

Let

X

bca Ba.nach space withaSchauderbasis

{b,}.

Let

{a,},k,n

q

N,

beanarrayof realconsiantssuchthat sup

[a,,k <

.

k

Let

{X,,},

1, 1,e a.sequence,,frandomelements in

X

which is

{a,}-compactly

CONVERGENCE IN MEAN OF WEIGHTED SUMS 447

T

hen:

limsup

la.lElIQ(X

EX)II"

O.

PROOF.

Given

>

0, there existsacompactsubset/t- of

X

such that

sup

E

Io.,lEIIXll"Zixm

<

2-2V(M +

1)-"

where

M

isthe basisconstantof the Schauderbasis

{b,

}.

Ve define,foreach n E

N"

W,,

t;,

X,,I[x.tq

X,,

W,.

The compactness of

It"

implies

(see

[71)

that there exists

to

E

N

suchthat

IIQ,(W,)ll <

c-

foreveryk

N

and

>

to,vhere

C

>

0 isaconstant suchthat sup

E

la’’:!

<

C.

k

Then,

for_evex3"k

N"

EIIQ,(Wk

EI4)]["

EllQ,(Wk)

<

2"-’

(EIIQ,(W)[I"

+

E"IIQ,(W)II) <_

2"EllQ,(W)I]"

<

2-"6

-On

the other hand:

k

<

2"(M

+

z)"

Io.,,IEIIYII"

<

s2-". k

Therefore,

forevery

>

0"

sup

Z

la.,,IEIIQ,(X

EX,)II" <

2"-(2

-’’

+

2

-v)

.

k

In

a simila,r manner, the folloving lemma,where the

restriction/ >

1 is omitted, canbe

proved:

LEMMA

4.2. Let

X,

{a,,}

and

{X,

bea.in

Lemma

1, with p

>

0. Then:

limsup

Z

la,IEIIQ(X)II"

o.

THEOREM

4.2. Let,

X

be a separableBana.ch space.

Let

{a,},k,n

N,

beanarray ()freal constantssatisfying:

a)

sup

Z

]a,,/,.

<

cx

b)

linsup

la.l

0.

k

Then

Ell

Z

a,,k(X.

EXk)]I

0 as n oc.

PROOF.

The

{a,,.

}-compactly

uniformintegrabilityof

{X, },

the consideration of bound-edness of thecon’esponding compact

K

and the condition

a)

yield

Z

I=,IEIIX

EX,, <

o

k

foreachn(

N,

andsothe almostsureconvergenceof

S,

for eachn(N.

Since

X

can be isometrically embedded in a Banach space with a Schauder basis, it can

be

assumed,

without lossofgenerality, that

X

has aSchauder bmis

{b,,};

let

M

bethe basis constant.

For

each fixed

N,

and for every

N"

Accordingto

Lemma

4.1, given

>

0, there exists

N

such that

E][Q,

Z

a,,(X,

EX,)

I1<

fo-.ve-y ,e2v.

Wefix sucha ]V: let m

1<i<

Thereexistsa compact subsetIfof

X

such that:

sup

Z

la"lEIIX*’ll-rtxzq

<

4m.t

We define,for each n

N:

W,,

X,,ltx,,etcl

Y;,

X.I[x.,.]

X.

W,,.

We

have:

f,(Wk- EWe,)},

k N,is,fl,reach

N,

asequence ofpairwise independent randomvariables withmeanO, and,consequently"

CONVERGENCE IN MEAN OF WEIGHTED SUMS 449

k

everyn

_> no

and 1

As

E

lan’12

-<

(sup

la,,

I)

E

la’kl

0 whenn o,we camchoose

no

E

N

such that for

On

the other hand:

Therefore

E

lib, _<

2m

E

la,,k]EI]Y]I-Ell

a.(X

EX)II <

+

2m...

+

e.

k

foreveryn

_>

no.

EXAMPLE

4.1. Thefolloving example

(suggested

byanotheronein

[9])

shows that the conditionsinTheorem 3.1, and therefore the

{a,k}-compactly

uniformp-thorderintegrability, areweaker thaz thecompactlyuniformp-th orderintegrability.

So,

ourTheorem 4.2isstrictly

,stronger

thanTheorem3.2in

[6]:

Consider theseparableBanach space and let

{e,

bethe standard basis.

Let

{X, },

n E

N,

be the sequence of independent random elements in definedby

he, withprobability

2,

X,

-he, withprobability

0 with probability 1

If

K

is any compact sult of

,

then

K

contains a.t mostfinitely manyelementsof the set

{=t=ne,,n e

g},

and so

supEIIX,,[lI[x.lq

1 whichimplies that

{X,,}

is not compactly

uniformlyintegrable.

Now,

given :

>

0 xve choose n0

N

such that 1

<

and let

K

{0,:kne,

n

0

1,2,...,n0}.

K

isa compactsubsetof

,

and

P[X,,

It’]=

l0

ifn_<n0

,

if

>

n0

Thus,

{X,

is tight.

Let

{ank}

beanm]ayofreal constants, andleta

>

n0.

Then,

foreveryn

N"

k

k

,,d o

{ilX.ll}

i

{a.}-unif,,nnly

integrableforany array

{a,,k}

such that sup

kl.l

<

;’

k

ill <lc

<n

for_ns_ance,

a.

Therefore,

{X,

is

{a,,k

}-conpactly

uniformlyintegra.blefor suchan array

{ank }.

Notethat the sequence

{X,

doesnotverify thehypothesisofcompactlyuniform integra-bility in

[6],

Theorem 3.2; neverthelessourTheorem 4.2 showsthat the thesis is truefor the

array

{ank

in theexample.

REMARK.

The definitions and results of this papercanbeformulatedbyconsideringan a.lTay

{X,k,

1

<

k

<_

k,

<

cxz,n

> 1}

ofrandomelements, and, basically, nothing wouldbe

changed

intheproofs.

We

havepreferredthe formulation forasequence

{X,,},n

E

N,

in order to stay within the framework of the classical

WLLN

and make easier the comparison ofour results withthe clmsicalonesin the literature.

For

recentsresultsonthe

WLLN

for arraysofrasadom variableswerefer to

Gut

([10])

and

Hong

andOh

([11]).

ACKNOWLEDGMENT.

The author is gratefifl to the referee for his comments and

his suggestions. This work is supportedin pa.rt by

DGICYT

grant PB93-0926 and

Junta

de Andalucla.

REFERENCES

[1]

ORDONEZ CABRERA.

M., "Convergence

of weightedsumsof random variablesand

uni-formintegrability concerningtheweights", Collect. Math. 45

(2) (1994),

121-132

[2]

CHANDRA,

T.K.,

"Uniform integrabilityin the Cesha’o sense and the weak law oflarge numbers",Sankhya.

Set.

A

51

(1989),

309-317

[3]

HOFFMANN-JORGENSEN,

J.

and

PISIER,

G.,

"The la.w oflarge numbersand the central limit theorem inBa.nch

spa.ces", Ann.

Probab. 4

(1976),

587-599.

[4]

DAFFER,

P.Z.

and

TAYLOR,

R.L.,

"Tightnessandstronglawsof

large

numbers inBanach

spaces",

Bull. In.,t. Math. Acad. Sin,ca 10

(1982),

251-263.

[5]

WANG,

X.C.

and

RAO,

M.B.,

"Some

results on the convergence of

weighted

sums of random elements inseparableBanach

spaces",

Studia Math. 86

(2) (1987),

131-153.

[6]

WANG, X.C., RAO,

M.B. and

DELI,

L., "Convergence

inther-thmeanofweightedsums

ofrandomelements in Banachspacesof type

p",

Northea.,tMath.

J.

8

(3) (1992),

349-356.

[7]

TAYLOR., R.L.,

"Stochastic

Convergence

of

Weighted

Sums

of

Random. ElementsinLinear

Spaces

",

Lecture

Notesin Ma.thematics, 672, Springer Berlin, 1978.

[8]

CUESTA,

J.A.

and

MATR/N,

C.,

"Strong

convergence of

weighted

sums of random

el-ements

through

the eqtfivalence ofsequences of distributions",

J.

Multivariate Anal. 25

(1988),

311-322.

[9]

ADLER, A., ROSALSKY,

A.and

TAYLOR, R.L.,

"Some

stronglaws of

large

numbers for

sumsof randomelements", Bull.

In.t.

Math. Acad. Sinica 20

(4) (1992),

335-357..

[10]

GUT, A.,

"The weaklawoflargenumbersfor

arrays", Star.

Probab. Left. 14

(1992),

49-52.

[11]

HONG, D.H.

and

OH, K.S.,

"On

theweak la.w of

large

numbersfor

arrays",

Star.

Probab.