Convergence in mean of weighted sums of {an,k}-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 of a veighted sum k a.k(Xk EXk) of random elements in a separable Banach space is studied under a new hypothesis which relates the random elements with their respective weights in the sum: the {a.. }-compactly uniform integrability of {X. }. This condition, which is implied by the tightness of {X,,} and the {a,,k }-uniform integrability of {[IX,, II}, is weaker than the compactly miform integrability of {X,,} and leads to a result of convergence in mean which is strictly stronger than a recent result of Wang, Rao and Deli. KEYWORDS AND PHRASES: Weighted sums, random elements in separable Banach spaces, compactly uniform integrability, {,,, }-compactly uniform integra.bility, tightness, {an, }- uniform integrability, convergence in mean. 1991 AMS SUBJECT CLASSIFICATION CODES: 60B12, 60F05. 1. INTRODUCTION. Let (f, A, 7’-) be a probability space, and let {X,, }, n N[, be a sequence of random elements in a. separable Ba.nach space (X, [[.[[), i.e., a sequence of flinctions from into Xwhich e A-metwa.ble with respect t() the Borel subsets of X. Let {a,,},k,n e N, be an array of real numbers with sup ]a,[ < . In this paper, we deal with the convergence in mean of the sequence of weighted sums a.(X. EX ). S,, Habitually, this problena of weak convergence, m well as the problem of strong convergence, ha been studied by considering epa,rately the c()nditi(ms on the sequence {X. (relying more and more the initial hyl)ot, h(.sis of independence mad identical distribution) and the conditions on the array {a,,. ). Ord6fiez ([1]) obtains esults ,,f convergence in mean and convergence in r-mean (r (0, 1)) for weighted sulns of random variables (random elements in ) by requiring a condition whi rela.tes the random variables .k’ to their respective weights a,,: the {a,, }-uniform integrability of {X,,}. This c,ndition is weaker than the uniform integrability of {X,} and leads to the 444 M. 0. CABRERA Ceshro uniform integra.bility (see [2]) as a particular case. A counterexample shows that the result of convergence in me,’m does not hold, a, sta.ted, for weighted sums of random elements in a separable Bana.ch space. In this paper, we obtain such a result, under a new condition on the random elements Xk concerning their respective weights in S,,: {X,,} is required to be {a,k }-compactly uniformly integrable. This requeriment is weaker than the requirement of compactly uniform integrability used by Hoffmann-.]orgensen and Pisier ([3]) and Dafter and Taylor ([4]), and characterized by Wang and Rao ([5]); consequently, our result extends the result obtained by Wang et al. ([6], Theorem 3.2). 2. PRELIMINARIES. In the proof of the main result, the embedding of a. separable B.’mach space X in a Banach space with a Schauder basis xvill be used as in Taylor ([7]). A sequence b. },. fi NI, in X is a Schauder basis for X if for each x X there exists a unique sequence of rea.l numl)ers {t. } such that z Z t.b.. When a Banach spa.cc X has a Sclmuder basis b. }, a sequence of continuous linear functionals {.f,, on X can be defined by f,,(x) t,,, n IN; these are called 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. (.:), for every x e X. U. and Q. axe two sequences of continuous linear operat,,rs on X satisfying lim U,,(x) x and lim Q.(x) 0 for every x fi X. A sequence {X,, of rndm elements in a Banach space X is said to be tight if for each e > 0 there exists a cnnpact subset K of X such that sup P [X,, t K] < e. Let p > 0. A sequence {X,, }, n G N, of random elements in a, Bh.nach space X is said to be compactly uniformly p-th orler integrnble if for every e > 0 there exists a compact subset K of X such that ,,p EllX, ll"Z[X.l,-] < :. (IA denotes the indicato: of the event A). If p 1, {X,, is said t, be compactly uniformly intcgrable. For the characteriztion ,,f this concept, we refer t,, rang and Rao ([5]) and Cuesta d Matrn ([8])" Let {.,, be a sequence ,,f rand,,m elements in a separable Bach space, d let p > 0. Then, {X,, is c,,mpactly uniformly p-th order integrable if, and only if, {X.} is tight and {[IX,,I]"} is uniformly integrable. For the relation bet:veen the notions of tightness, uniform integrability, boundedness of xnoments and domina.tion in probability, we refer to Taylor ([7]). CONVERGENCE IN MEAN OF WEIGHTED SUMS 445 3. COMPACTLY UNIFORMLY INTEGRABLE RANDOM ELEMENTS CONCERNING AN ARRAY. Ord6fiez ([1]) introduces the folloving 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,} of integrable random viables is said to be {a,}-unifoly integable (or uniformly int.egrable concerning the axray a }) if lim sup [o,. [E[X [I[Ix I>,,] O. The following msertion is easy to check: PROPOSITION 3.1. Let {X,,} be a sequence of uniformly integrable rdom vabl. Then, {X,,} is {a,, }-uniformly integrable for all m’ra.ys {a,,k sud that sup [ak [< . k A sequence of rmdom elements in a separable Bana.ch space being compactly uniformly integrable is the natural extension of a. sequence of random variables being uniformly integrable (both definitions are equiva.lent when the Ba.nach spa.ce is finite dimensional). To this effect, we introduce the following notion" 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 a sepable Banach space X is said to be {a,,a }-cmapactly uniformly p-th order integrable if for eve > 0 there exists a. compact subset K of X such that k If p 1, {X,, is sa.id t,, be {a,, }-compactly uniformly integrable. The following theorem provides a sufficient condition ff, r the {a,, }-compactly uform p-th order integrability of 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, be a sequence of rdom elements in a se arable Sanach space X, which is tight, and such that the sequence {[[X,,[[ v} is {an}-unifoly integrable. Then, {X,, is {a,, }-c,,xnpa,’tly mfiformly p-th order integrable. PROOF. By Theoen 2 in [1], given > 0, there exists > 0 such that whenever {Ak} is a sequence ,.,f events satisfying sup ]a,,IP(A) < 6, then sup [a,,[E[[X[["IA, < e. kk 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 > 0 is any constant such tha.t sup [a,[ C. k Then sup [",kiP [X I(] < $, and therefore: k 446 M.O. CABRERA i.e., {X,, is {a,,k }-compactly uniformly v-th order integra.ble. It is easy to check the following PROPOSITION 3.2. Let {X, be a sequence of compactly uniformly p-th order (p > 0) integrable random elements in a. separable Banach space. Then, {X,, is as in Theorem 3.1, and so {X, is {a,,k }-c,mpactly uniformly p-th order integrable for all arrays {a,, } such that sup E k REMARK. The characterization of {a,, }-compactly unifo.rn p-th order integrability of X,, in terms of tightness of {X,, a.nd a,, }-uniform integrability of {ilX,, ’ is not available (i.e. the condition in Theoren 3.1 is not necessary), as is shown by considering the sequence of rmdom elements in {.’ I" II.’ll E Ix,,I < oo} defined by X,, e,, with probability 1, where e, } is the standard basis of ,’rod the ,’u’ra.y ifl<k<n a,,= 0 ifk>n. Givene>0, taken ENsuchtha.t & <,andlet the compact K={e e2, ,era} Then k > Therefore, {X,,} is {a,,t.}-compactly uniformly p-th order integra.ble (p > 0), but {X,,} is not tight. 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, be a.n array of real constants satifying: k b) hm sup 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.n extension is possible for a sequence of raadom elements {X,, being {a,,}-c(,mpactly unifornfly integra.ble. Previously, we prove the fl)llowing lemma" LEMMA 4.1. Let X bc a Ba.nach space with a Schauder basis {b,}. Let {a,},k,n q N, be an array of real consiants such that sup [a,,k < . k Let {X,,}, 1, 1,e a. sequence ,,f random elements in X which is {a,}-compactly unifi,rml p-th order integrable flr some p 1. CONVERGENCE IN MEAN OF WEIGHTED SUMS 447 T hen: limsup la.lElIQ(X EX)II" O. PROOF. Given > 0, there exists a compact subset/t- of X such that sup E Io.,lEIIXll"Zixm < 2-2V(M + 1)-" where M is the basis constant of the Schauder basis {b, }. Ve define, for each n E N" W,, t;, X,,I[x.tq X,, W,. The compactness of It" implies (see [71) that there exists to E N such that IIQ,(W,)ll < cfor every k N and > to, vhere C > 0 is a constant such that 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, for every > 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, can be proved: LEMMA 4.2. Let X, {a,,} and {X, be a. in Lemma 1, with p > 0. Then: limsup Z la,IEIIQ(X)II" o. THEOREM 4.2. Let, X be a separable Bana.ch space. Let {a,},k,n N, be an array ()f real constants satisfying: a) sup Z ]a,,/,. < cx b) linsup la.l 0. k Let {X }, n N, be a, sequence ,,f pairwise independent and {a, }-compactly ifoly integrable random elem,nts in X. 448 M.O. CABRERA Then Ell Z a,,k(X. EXk)]I 0 as n oc. PROOF. The {a,,. }-compactly uniform integrability of {X, }, the consideration of boundedness of the con’esponding compact K and the condition a) yield Z I=,IEIIX EX,, < o k for each n ( N, and so the almost sure convergence of S, for each n ( N. Since X can be isometrically embedded in a Banach space with a Schauder basis, it can be assumed, without loss of generality, that X has a Schauder bmis {b,,}; let M be the basis constant. For each fixed N, and for every N" According to Lemma 4.1, given > 0, there exists N such that E][Q, Z a,,(X, EX,) I1< fo-.ve-y ,e2v. We fix such a ]V: let m 1<i< There exists a compact subset If of 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,r each N, a sequence of pairwise independent random variables with mean O, and, consequently" E . o.,,(W. EWe) Ile,,ll E1/2 Z a,,,.f(W EWe,) by the boundedness of K (A is a constant). CONVERGENCE IN MEAN OF WEIGHTED SUMS 449 k every n _> no and 1 As E lan’12 -< (sup la,, I) E la’kl 0 when n o, we cam choose 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 for every n _> no. EXAMPLE 4.1. The folloving example (suggested by another one in [9]) shows that the conditions in Theorem 3.1, and therefore the {a,k}-compactly uniform p-th order integrability, are weaker thaz the compactly uniform p-th order integrability. So, our Theorem 4.2 is strictly ,stronger than Theorem 3.2 in [6]: Consider the separable Banach space and let {e, be the standard basis. Let {X, }, n E N, be the sequence of independent random elements in defined by he, with probability 2, X, -he, with probability 0 with probability 1 If K is any compact sult of , then K contains a.t most finitely many elements of the set {=t=ne,,n e g}, and so supEIIX,,[lI[x.lq 1 which implies that {X,,} is not compactly uniformly integrable. Now, given : > 0 xve choose n0 N such that 1 < and let K {0,:kne, n 0 1,2,...,n0}. K is a compact subset of , and P[X,, It’]= l0 ifn_<n0 , if > n0 Thus, {X, is tight. Let {ank} be an m]ay of real constants, and let a > n0. Then, for every n N" k k ,,d o {ilX.ll} i {a.}-unif,,nnly integrable for any array {a,,k} such that sup kl.l < ;’ k ill <lc <n for _ns_ance, a. 0 if 1," > . 450 M. 0. CABRERA Therefore, {X, is {a,,k }-conpactly uniformly integra.ble for such an array {ank }. Note that the sequence {X, does not verify the hypothesis of compactly uniform integrability in [6], Theorem 3.2; nevertheless our Theorem 4.2 shows that the thesis is true for the array {ank in the example. REMARK. The definitions and results of this paper can be formulated by considering an a.lTay {X,k, 1 < k <_ k, < cxz, n > 1} of random elements, and, basically, nothing would be changed in the proofs. We have preferred the formulation for a sequence {X,,},n E N, in order to stay within the framework of the classical WLLN and make easier the comparison of our results with the clmsical ones in the literature. For recents results on the WLLN for arrays of rasadom variables we refer to Gut ([10]) and Hong and Oh ([11]). ACKNOWLEDGMENT. The author is gratefifl to the referee for his comments and his suggestions. This work is supported in pa.rt by DGICYT grant PB93-0926 and Junta de Andalucla. REFERENCES [1] ORDONEZ CABRERA. M., "Convergence of weighted sums of random variables and uniform integrability concerning the weights", Collect. Math. 45 (2) (1994), 121-132 [2] CHANDRA, T.K., "Uniform integrability in the Cesha’o sense and the weak law of large numbers", Sankhya. Set. A 51 (1989), 309-317 [3] HOFFMANN-JORGENSEN, J. and PISIER, G., "The la.w of large numbers and the central limit theorem in Ba.nch spa.ces", Ann. Probab. 4 (1976), 587-599. [4] DAFFER, P.Z. and TAYLOR, R.L., "Tightness and strong laws of large numbers in Banach 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 in separable Banach spaces", Studia Math. 86 (2) (1987), 131-153. [6] WANG, X.C., RAO, M.B. and DELI, L., "Convergence in the r-th mean of weighted sums of random elements in Banach spaces of type p", Northea.,t Math. J. 8 (3) (1992), 349-356. [7] TAYLOR., R.L., "Stochastic Convergence of Weighted Sums of Random. Elements in Linear Spaces ", Lecture Notes in Ma.thematics, 672, Springer Berlin, 1978. [8] CUESTA, J.A. and MATR/N, C., "Strong convergence of weighted sums of random elements through the eqtfivalence of sequences of distributions", J. Multivariate Anal. 25 (1988), 311-322. [9] ADLER, A., ROSALSKY, A. and TAYLOR, R.L., "Some strong laws of large numbers for sums of random elements", Bull. In.t. Math. Acad. Sinica 20 (4) (1992), 335-357.. [10] GUT, A., "The weak law of large numbers for arrays", Star. Probab. Left. 14 (1992), 49-52. [11] HONG, D.H. and OH, K.S., "On the weak la.w of large numbers for arrays", Star. Probab. Left. 22 (1995), 55-57. 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