Internat. J. Math. & Math. Sci. VOL. 18 NO. (1995) 33-36 33 ON COMPLETE CONVERGENCE OF THE SUM OF A RANDOM NUMBER OF STABLE TYPE P RANDOM ELEMENTS ANDRI ADLER Department of Iathelnatics lllinfis Institute of Technology Chicago, IL 60616 U.S.A. ANDREY VOLODIN Research Institute of Mathematics Kazan University Kazan Tatarstan Russia 420008 (Received September 1, 1992 and in revised form March 10, 1993) ABSTRACT. Complete convergence for randomly indexed normalized sums of random elements of the form T=. X)/O(T,,)is established. The random elements {X,} belong to a type p stable space and m e assumed to be independent, but not necessarily identically distributed. No assumptions are placed on the joint distributions of the stopping times {T.}. KEY WORDS AND PHRASES. Complete convergence, stable type p. 1990 AMS SUBJECT CLASSIFICATION CODES. 60F15, 60B12. In this article we extend previous results on complete convergence for randomly stopped sums. A sequence of random elements {X, is said to converge completely to zero if for all e > 0 r--i In view of the Borel-Cantelli lemma it s clear that complete convergence implies almost sure convergence. Hsu and Robbins [15] are credited with the introduction of this concept. Since then many well known mathematicians have explored this interesting subject. This paper generalizes the work of Adler [1] and Gut [3] via methods that can be found in Volodin [7]. We extend both articles in a couple of ways. First of all our random variables are no longer solely defined on the real line. Also, we introduce the function (x), which we use to normalize the partial sums. Hence, the norming sequence is not just the number of terms in our partial sum, but a function of the number of terms. One should compare Theorem 1 with Theorem 4.1 of Gut [3] and Theorem 2 with Theorem 2 of Adler The first definition that we need to introduce is that of a stable type p Banach space. Let {,} be a sequence of independent random variables with characteristic function exp{-itl}, where 1 < p < 2. The Banach space E is said to be of stable type p if
34 A. ADLER AND A. VOLODIN -],,=-y,,x,,. Set p(e) s,q{p E is of st.a]fl, type p}. N,t, that ince the interval of st])h tylws is len and p < 2 we cn select num])cr r such that p < r < p(e). Insteal f the usual hyl),th,sis, that nu" rmdom cl mcnts all have the same distributiara, w, assume, that there is a dominatink radnn varia]h, i.e., xw say that X dominates {X, if there xists a constant D s, that,,pp{]lx,,ll>t} DP{X >t} for,llt >0. Next, let {,,} b. a sequence of strictly increasing integers. Also, let {(n)} be a squcnce of positive constants such that () aa/ for all n and some positive constant (,. AS IISU Set ]I(t)= ZI ", (t)= card{,, "a, t} and Set,, Y=l X. Finally, note that the constant C is a generic constant which is not necessarily the stone in each appearance. In most situations we e only concerned with obtaining upper bounds to our terms. Hence when one term is majorized by another we combine all the coefficients into one, which we denote by C. Our first result exhibits a classical strong law. It is classical in the sense that the stopping times are not random. It will be of great value in proving Theorem 2. THEOREM I. Let E be a Banach space of stable type p, where p < 2. Let {X,} be a sequence of independence mean zero random elements in E. Let X be a positive rdom viable that dominates {X,}. If EX" <, where p < r < p(e) and EM(O((2X))) <, th S./(.) ovg ompety to zro. PROOF. Without loss of genermity we may sume that {Xn is a sequence of symmetric random elements. Since r > p we can select a nber m so that (r/p-1)2 one. By reapplying Hoffmann-Jorgensen s [4] and Chebyshev s inequmities we have for exceeds Z P{IIS,,II > e3"(a,,)} _< C Z[P{IIS,. [I > n--1 n- l + C max P{llx, II > _< c k--(.)lls. + C a,,p{x > _< C,[+-(.1 n=l 2-"1 +CEM(((X))) <_ C(EX") Z[a,-r(a, 1]" + 0(1) n-i
CONVERGENCE OF THE SUM OF A RANDOM NUMBER 35 < C -( h - )2" + 0(1 o(), which completes the proof. Next ve introduce our stopping times. Let {Tn} be a sequence of integer valued random variables. The ensuing theorem establishes a strong law for a randomly stopped SUlll. THEOREM 2. In addition to the hypotheses of Theorem suppose that {fl,,} is a sequence of positive constants with limsupn_ fin < 1, -n,/3.4y-r(an) < c and -,=1P{IT./a. 11 > 3-} < oo. If 4>(2a.) < bc(a.) for all n and some constant b, then ST,,/c(T.) converges completely to zero. PROOF. In view of Theorem we may conclude that So,,./ck(a,) converges completely to zero. Next, we show that (ST,. So.)/$(a,,) converges to zero, which implies that $7". / b(a,,) converges completely to zero. Set A. {k [a,,(1-fl.)]+l _< k _< a.-1} andb. {k a,,+l _< k _< [a.(l+fl.)]}. Let > 0. By applying Etemadi s [2] maximal inequality and Cheyshev s inequality we have P{llSz, s,, II > 4,(,,.,)} n--1 +ZP{ --T"-I >/3.} n=l
36 A. ADLER AND A. VOLODIN < n-- 0(I). Fro, n the hypothesis (2c,) < be(a,,)it follows that for all 0 < b < 1" ()c,) > C(a,,). Next, choose fl and N so that/3. </3 < 1 for all n >_ N. Observe that n=l P{IIST. II > (T,,)} P{IIST. II >,(T.)} + 0(1) which completes the proof. < Y P{IISToll > e(t.), T. 1 _< + E P{ T, _1 >.}+O(1) n < _, P{IIST.II > (T.),T. >.(1 )} + O(1) < P{IIST.II > (c,(1 fl))} + 0(I) < Y P{llSr.II > c(.)} + O(1) O(1), For more on recent results on complete convergence for Banach valued random elements that also explores the issue of randomly indexed sums see Kuczmaszewska and Szynal [6]. REFERENCES 1. ADLER, A. On complete convergence of the sum of a random number of random variables, Calcutta Statist. Assoc. Bull. 37 (1988), 161-169. 2. ETEMADI, N. On some classical results in probability theory, Sankhya ser. A 47 (1983), 215-221. 3. GUT, A. On complete convergence in the law of large numbers for subsequences, Ann Probab. 13 (1985), 1286-1291. 4. HOFFMANN-JORGENSEN, J. Sums of independent Banach space valued random variables, Studia Math. 11 (1974), 159-186. 5. HSU, P.L. and ROBBINS, H. Complete convergence and the law of large numbers, Proc. Nat. Acad. Sci. U.S.A. 33 (1947), 25-31. 6. KUCZMASZEWSKA, A. and SZYNAL, D. On complete convergence in a Banach space, Inter. J. Math. and Math. Sci. 16 (1993). 7. VOLODIN, A. On Be-convex spaces, Izvestia VUZ Mathematica 4 (1992), 3-12.
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