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Internat. J. Math. & Math. Sci.VOL. 21 NO. 4 (1998) 749-754

79

COMPLETE CONVERGENCE FOR B-VALUED L’-MIXINGALE SEQUENCES

UANG HANYING and REN YAOFENG

Department of Statistics and Finance

University of Science and Technology of ChinaHefci,Anhui 430072, P.R.China

andThe Naval Academic Institute of Engineering of China

Wuhan 430033, P.R. China

(Received April 19, 1996 and in revised form January i0, 1997)

ABSTRACT. Under weaker conditions of probability,we discuss in this paper the complete con-

vergence for the partial sums and the randomly indexed partial sums of B-valued /)’-mixingale

sequences.

KEY WORDS AND PHRASES" Complete convergence,LV-mixingale sequence, q-smooth

Ianach space.

1991 AMS SUBJECT CLASSIFICATION CODES: 60F1560B12.

1. INTRODUCTION AND MAIN RESULTSSince the definition of complete convergence for real random variables was introduced by ttsu

and RobbinsIll,there have been an extensive literature in the complete convergence for indepen-

dent and dependent random sequences,see partially the references listed.In particular,Yang[5,6ha. discussed the complete convergence for B-valued independent random elements.Yu[ll] ha

co,sidered the complete convergence for martingale difference sequences, PeligradI7 and Shao[S}have obtained the complete convergence for C-mixing sequences,respectively.tlowever,to our best

acknowledgement,there are still few articles on the complete convergence for L-mixingale (1 _<:p < 2) sequences,which include uniformly rnixing (called also -mix;ing) sequences,martingale

difference sequences,linear processes and other random sequences (see I0]).In this paper,under

wakcr conditions of probability,we discuss the complete convergence for the partial sums and

the randomly indexed partial sums of B-valued L-mixingale sequences, and give the complete

convergence for B-valued martingale difference sequences as corollary.The methods used here are

different from those used in the literature.

Next,let us introduce some notations.

Let B be a real Banach space.B is said to be q-smooth(1 < q _< 2) if there exists a constant

C > 0 such that for every B-valued L -integrable martingale difference sequence (D," > 1

Ell D, II’ < C, EIID, ", , > ,.lct {X,,n > l} be a sequence of B-valued L -intcgrable (1 < p < 2) random variables on

a probability space (I,.,P), and let {Y’,,-cx < n < o} be an increasing sequence of sub a-

field of Y.Then {X,, Y,} is called a LV-mixingale sequence if there exist sequences of nonnegative

constants C, and (n),where (rn) 0 as m ov ,which satisfy following properties"

50 L. HANYIIqG AND R. YAOFENG

(i) IIE(XIT_.,)I, <_ (m)c and

(ii) IIx,,- (x,,l:r,,,,,,)ll, _< O(,r, +for all n > and m > O,where IlXll, (EIIXll) ’/".

S is the class of all positive non-decreasing function on R+ [0, o)(see [9],p.228 or [5,6])satisfying the following conditions:

(i) There exists a constant, k k() > 0 such that

(=v) _< ((=) + ()), v=, v n+.

(ii) z/qb(z) is non-decreasing for sufficiently large x.

From now on ,we will use C to denote finite positive constants whose value may change from

statement to statement .For real number z, Ixl denote the largest integer k _< x.I(A) represent

indicative function of set A. Put S X,.

THeOreM 1.1. bet < q2,0< < - 1,1 p 2, d= or-l,and let B be a

q-smooth Banaeh space.Supppose {X,} is a B-valued L-mixingale sequenee,() S.If

e(llX, ll’(e(llX, ll)) - > =) c=-’’ (.)

for sufficiently large x,n and there exists a X(1 A p) such that + (1 t)A > 0 and

([n’l) mx C < (1.2)l<t<n

q-twhere 0 < < ii (i,then for every > 0 we have

lp(m IISll > (n(())) ’/’) < (.3)

in particular - P(IIS.II _> (,((,))’)’/’) < oo. (1.4)n=l

/2

THEOREM 1.2. Under the assumptions of THEOIEM 1.l,if there exists a A(1 A p)satisfying

m2([2"]) max C < , (1.5)

then for every > 0 we have

-1p(up[ll&ll/([(k)))’/’ )< . (1.6)n=l

If {X,,,n 1} is a -integrable B-valued martingale difference sequence,then C(EIIXII) v, (m) 0 for m 1.

COROLLARY 1.1. Let < q 2,d or-l,and let B be a q-smooth Banach

space.Supppe {X,, ,,n 1} is a B-valued martingale difference sequence,(z) S.For 0 << and suciently large z and n,if (1.1) is satisfied,then for every > 0, we obtain that

( .z),(.4) ,a (.6)RMARK 1.1. For 0 < < 1,by C-inequality and properties of (z), we can prove

hal, the results of THEOREM 1.1,THEOREM 1.2 and COROLLARY 1.1 hold for y B-valuedrando,n variable sequence {X,,n 1} wihou mixing condition (1.2) and (1.5).

REMARK 1.2. Real uniformly mixing sequence (definition see [7] or [s],[x01) (x, ,} i

L -mixingale,where C, 2(EX)/,(m)= /(m),see [0,p.9].

COMPLETE CONVERGENCE FOR THE PARTIAL SUMS 751

REMARK 1.3. YangI51 has proved that (1.3) and (1.4) hoM for B-valued independent

zero mean random element sequence {X.} in type 2 Banach space under moment conditionsstronger than the conditions of COROLLARY 1.1.

2. PROOFS OF MAIN RESULTS

We only prove the case in d for Theorem 1.1 and 1.2,the proof of the case in d -1 is

analogous.

LEMMA 2.1.([9],Lemma 1) Let (.) E S, 6 > 0,then for any z > 0,

c(=) < (=(=)) <

c() <_ (/()) < cc);

c() < () _< c().

PROOF OF THEOREM 1.1. Notice first

Obviously

-P(max IISll >n=l l<k<rt

_< .=,E -P(,<<.max IIA,II >_ ’ (=(=))’/’).:, ;p(,<m<x. IIcll _> (())’/’)

g I+Is+13.the Markovian inequality,/)’-mixingale property and the properties of (x),we have

, _< c (,,(.//-#(<_ c (())-/’(: IIX,- ECX, l,+[..])ll)

n=l =1

max C,"-1

< C 2 (InoI) maxrt=l l<<n

Similarly,we can obtain

13 < C (lntl) max C < oo.l_<,<n

< I,iClearly,X, Y,,. + Z,,.,Wt, U,, + V,,.For fixed t, (U,,E+, _< _< n} and {V,,,.,+t, <

752 L. HANYING AND R. YAOFENG

__< n} are martingale difference sequences. Then

l=-[nPl+l l<k<n t=l

g I + I.Since B is q-smoothable,therefore using Doob inequality,the monotone property

and iemma 2.1 we have

n=l l=-[nP]+ t=l

n=i

n=l =1

By applying the definition of (x) and Lemma 2.1,we can obtain

e() ce (2.)

for ,/ > 0 and sufficiently large z.

By the Markovian inequality,the definition of $(z),Lemma 2.1 and (2.1) we have

The proof is completed.

PROOF OF THEOREM 1.2. First,by the monotone property of (x) we have

1’(sup(ll&lll(k(k))’/’) >__n=l k>n

<_ P(sup(llSll/(k(k))’/’) >_,=0

< C E raP( max (llS, >m=l 2"5k<2

Observe that for 2 <_ k < 2"+,

,s’ E(x,-t"-I

iv""l+ E E (E(x, 13;,+,)-

=-12""1+1 ,=’

+ E

COMPLETE CONVERGENCE FOR THE PARTIAL SUMS 753

Then

kpCsup(lls, ll/((,)) ’#) >_ )n=l k>n

< C mP( max (IIAII > (2())’/’)m=l m+

+C E raP( max (IIBII >5+c fmP(.. <<-+,max ([[c[[ >_ 5’(2m(2m)) ’/’)I+Ir+I.

fly analogizing the proof of I we have

I6 _< C m:Z’f([2’]) max C, < oo.m=l

By analogizing the proof of Is,similarly, we can obtain 18 < oo.

Let Y,.,, X,/(llX, <_ (2"’(2"))’/t),Z,.,, X,-Y,.,.,,,W., E(X,[+,)-E(XIT,+,_,),U,,-E(Y,,.IZ+,)- E(Y,.,IZ+,_,),V.,--- E(Z,,..IZ+,)- E(Z,,,.,,IZ+,_,),2" < k < 2"+’,, < <k, -121 +, _< _< [2a’l. Then

I7 < C m E P(.. maxm=l I=-[2"]41 <k<2m+t *=1

+C E m E P( max E v,,,ll > (2"(’)) ’/’- 2-)m=l l=-[2a’]+l 2" <k< 2"4 t=l

19 + 1,0.

ly analogizing the proof of h we have

19 _<_ C m2(a+qa+’-q/O((2"’))-q/t + C m2(--’)" < .llv analogizing the pr-of of Is we have

1,o <__ C m2(-’) + C m2("-‘) j,[((yt)),+tS. RANDOMLY INDEXED PARTIAL SUMS

Throughout this section let {X,,7,} be a B-valued LP-mixingale sequence (l p 2),and let {r,, n 1} be a sequence of nonnegative,integer valued random variables.r is a positive

random variable.All random variables are defined on the same probaSility space.

THEOREM 8.1. Under he sumptions of THEOREM 1.2,if there exists some constant

to > 0 such that ,(3.1)-e(-- < ’0)< ,

n=l

then for every > 0 we have ., 1p(llS,.I >_ ,(.((.)))’/’) < oo. (3.2)rt

THEOREM 8.2. Under the assumptions of THEOREM 1.1,if there exist constants a, b, t0(0 <a_<b<oo) such that P(a _< r _< b) and

P(I-- *1 > ,0) <n=l

then for every > 0 we obtain that (3.2) holds.

754 L. HANYING AND R. YAOFENG

THEOREM 3.3. Under the assumptions o[ THEOREM 1.l,i[ there exist constants b > 0

and e0 > 0 such that P(r _< b) and (3.3) is satisfied,then for every > 0 we have

[P(llS,.,,ll > (((,,))’)’/’) <n=l n

Obviously,suppose P(r > a) 1 for some a > 0,then for any > 0(e < a) we have

p(r,, < a e) .<_P(l

therefore,if condition (3.3),where P(r _> a) for some a > e0 > 0 replaces condition (3.1},thenTIIEOREM 3.1 still holds.

Similarly,using COROLLARY l.l,we can obtain the complete convergence for the randomly

indexed partial sums of B-valued martingale difference sequences,respectively.

REMARK 3.1. Condition (3.1) and (3.3) are just ones which are usually employed in

literature.

REMARK 3.2. Note that if (x) 1,Lk(x)(Lo(x) max(1,1og x),L log[max(e, Lk_,(x)],k 1,2,.--),we can derive many significative results from the results of this paper.In addition,

since real space is a 2-smooth Banach space,the TttEOREM and COROLLARY in this paper are

sitable for real valued random variable.

ACKNOWLEDGMENTS. The authors would like to thank Prof.Hu Dihe for his suggestions

and encouragement.In addition,they are very grateful to the referee and the editor for valuable

suggestions and comments.

REFERENCES

Ill IISl],P.L.and ROBBINS,H., Complete convergence and the law of large numbers, Pro. Na.Acad. Sci. U.S.A. 33(2)(1947),25-31.

[2] BAUM,L.E.and KATZ,M., Convergence rates in the law of large numbers, Trans. Arner.Math. Soc. 120(1)(1965),108--123.

[3] G UT,A., Complete convergence and convergence rates of randomly indexed partial sum,s withan application to some first passage time, Acta. Math. Hung. 42(3-4)(1983),225-232.

[41 BAI,Z.D.and SU,C., The complete convergence for partial sums of i.i.d.random vari-ables, Sc,ent,a Sinica(SeriesA),XXVIII(1985), 1261-1277.

[5] YANG,X.Y., Complete convergence of a claas of independent B-valued random ele-ments, Acta. Math. Sinica. 36(6)(1993),817-825.

16] YANG,X.Y., A note on convergence rates for sequences of B-valued randomly indexed partial"sums, Chinese Ann. Math. 14A(3)(1993), 275-282.

[7] I’ELIGRAD,M., Convergence rates of the strong law of stationary mixing se-

quences, Z. Wahr. Verw. Gebiete. 70(1985),307-314.[8] SliAO,Q.M., A moment inequality and its applications, Acta. Math. Sinica. 31(1988),736-

747.

[9] WU,Z.Q.,WANG,X.C.and LI,D.L., Some general results of the law of large num-

bers, Northeastern Math. J. 3(1987),228-238.[10] liALL,P.and HEYDE,C.C., Martingale Limit Theory and its Application, New

York,London,Academic Press 1980.

I11] YU,K.F., Complete convergence of weighted sums of martingale differences, J. Theor. Prob-ab. 3(1990),319-347.

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