A complete convergence theorem for arrays of rowwise independent
random variables was proved by Sung, Volodin, and Hu. In this paper, we extend this theorem to the Banach space without any geometric assumptions on the underlying Banach space. Our theorem also improves some known results from the literature.Korea Research Foundation Gran

Hu, Tien Chung

Ordóñez Cabrera, Manuel Hilario

Sung, Soo Hak

English

idUS. Depósito de Investigación Universidad de Sevilla

J. Korean Math. Soc. 44 (2007), No. 2, pp. 467–476ON COMPLETE CONVERGENCE FOR ARRAYS OFROWWISE INDEPENDENT RANDOM ELEMENTSSoo Hak Sung, Manuel Ordóñez Cabrera, and Tien-Chung HuReprinted from theJournal of the Korean Mathematical SocietyVol. 44, No. 2, March 2007c©2007 The Korean Mathematical SocietyJ. Korean Math. Soc. 44 (2007), No. 2, pp. 467–476ON COMPLETE CONVERGENCE FOR ARRAYS OFROWWISE INDEPENDENT RANDOM ELEMENTSSoo Hak Sung, Manuel Ordóñez Cabrera, and Tien-Chung HuAbstract. A complete convergence theorem for arrays of rowwise inde-pendent random variables was proved by Sung, Volodin, and Hu [14]. Inthis paper, we extend this theorem to the Banach space without any geo-metric assumptions on the underlying Banach space. Our theorem alsoimproves some known results from the literature.1. IntroductionThe concept of complete convergence of a sequence of random variables wasintroduced by Hsu and Robbins [5] as follows. A sequence {Un, n ≥ 1} ofrandom variables converges completely to the constant θ if∞∑n=1P (|Un − θ| > ε) < ∞ for all ε > 0.In view of the Borel-Cantelli lemma, this implies that Un → θ almost surely.The converse is true if {Un, n ≥ 1} are independent random variables. Hsu andRobbins [5] proved that the sequence of arithmetic means of independent andidentically distributed random variables converges completely to the expectedvalue if the variance of the summands is finite. Erdös [2] proved the converse.We refer to Gut [3] for a survey on results on complete convergence related tostrong laws and published before the nineties.The result of Hsu-Robbins-Erdös has been generalized and extended in sev-eral directions. Some of these generalizations are in a Banach space setting.A sequence of Banach space valued random elements is said to converge com-pletely to the 0 element of the Banach space if the corresponding sequence ofnorms converges completely to 0.Recently, Sung et al. [14] proved the following complete convergence theoremfor arrays of rowwise independent random variables.Received December 17, 2005.2000 Mathematics Subject Classification. 60B12, 60F15, 60F05.Key words and phrases. Banach space valued random elements, complete convergence,rowwise independence, sums of independent random elements, convergence in probability.This work was supported by the Korea Research Foundation Grant (KRF-2004-041-C00050).c©2007 The Korean Mathematical Society467468 SOO HAK SUNG, MANUEL ORDÓÑEZ CABRERA, AND TIEN-CHUNG HUTheorem 1. Let {Xni, 1 ≤ i ≤ kn, n ≥ 1} be an array of rowwise independentrandom variables, {kn, n ≥ 1} a sequence of positive integers and {an, n ≥ 1}a sequence of positive constants. Suppose that for every ε > 0 and some δ > 0:(i)∑∞n=1 an∑kni=1 P (|Xni| > ε) < ∞,(ii) there exists J ≥ 2 such that∞∑n=1an( kn∑i=1EX2niI(|Xni| ≤ δ))J< ∞,(iii)∑kni=1 EXniI(|Xni| ≤ δ) → 0 as n →∞.Then∑∞n=1 anP (|∑kni=1 Xni| > ε) < ∞ for all ε > 0.Theorem 1 was first presented by Hu et al. [8]. Hu and Volodin [9] imposedone additional condition in addendum to the paper. Many people tried to proveTheorem 1 without the additional condition (for random variables, see Hu etal. [6] and Kuczmaszewska [11], and for random elements, see Hu et al. [7]).The following theorem is a version of Banach space setting of Theorem 1and is due to Hu et al. [7]. No assumptions are made concerning the geometryof the underlying Banach space.Theorem 2. Let {Xni, 1 ≤ i ≤ kn, n ≥ 1} be an array of rowwise independentrandom elements, {kn, n ≥ 1} a sequence of positive integers and {an, n ≥ 1}a sequence of positive constants. Suppose that for every ε > 0 and some δ > 0:(i)∑∞n=1 an∑kni=1 P (||Xni|| > ε) < ∞,(ii) there exists J ≥ 2 such that∞∑n=1an( kn∑i=1E||Xni||2I(||Xni|| ≤ δ))J< ∞,(iii)∑kni=1 P (||Xni|| > δ) = o(1),(iv) ||∑kni=1 Xni|| → 0 in probability.Then∑∞n=1 anP (||∑kni=1 Xni|| > ε) < ∞ for all ε > 0.In this paper, we extend Theorem 1 to the Banach space without any geo-metric assumptions on the underlying Banach space. Our result also improvesTheorem 2. More precisely, Theorem 2 holds without condition (iii).We state our first theorem which shows that o(1) in condition (iii) of Theo-rem 2 can be replaced by O(1). The proof will appear in Section 3.Theorem 3. Let {Xni, 1 ≤ i ≤ kn, n ≥ 1} be an array of rowwise independentrandom elements, {kn, n ≥ 1} a sequence of positive integers and {an, n ≥ 1}a sequence of positive constants. Suppose that for every ε > 0 and some δ > 0:(i)∑∞n=1 an∑kni=1 P (||Xni|| > ε) < ∞,ON COMPLETE CONVERGENCE FOR ARRAYS 469(ii) there exists J ≥ 2 such that∞∑n=1an( kn∑i=1E||Xni||2I(||Xni|| ≤ δ))J< ∞,(iii)∑kni=1 P (||Xni|| > δ) = O(1),(iv) ||∑kni=1 Xni|| → 0 in probability.Then∑∞n=1 anP (||∑kni=1 Xni|| > ε) < ∞ for all ε > 0.The following theorem is our main result which shows that condition (iii) ofTheorem 2 can be removed. The proof will appear in Section 3.Theorem 4. Let {Xni, 1 ≤ i ≤ kn, n ≥ 1} be an array of rowwise independentrandom elements, {kn, n ≥ 1} a sequence of positive integers and {an, n ≥ 1}a sequence of positive constants. Suppose that for every ε > 0 and some δ > 0:(i)∑∞n=1 an∑kni=1 P (||Xni|| > ε) < ∞,(ii) there exists J ≥ 2 such that∞∑n=1an( kn∑i=1E||Xni||2I(||Xni|| ≤ δ))J< ∞,(iii) ||∑kni=1 Xni|| → 0 in probability.Then∑∞n=1 anP (||∑kni=1 Xni|| > ε) < ∞ for all ε > 0.As an application of Theorem 4, we have the following corollary, which willbe proved in Section 3.Corollary 1. Let {Xni, 1 ≤ i ≤ n, n ≥ 1} be an array of rowwise independentrandom elements which are weakly mean dominated by a random variable X,that is, there exists a constant C > 0 such that 1n∑ni=1 P (||Xni|| > x) ≤CP (|X| > x) for all x ≥ 0 and n ≥ 1. Let α and φ be nondecreasing functionsdefined on (0,∞) satisfying0 < α(x) ↑ ∞ and 0 < φ(2x) ≤ Dφ(x) for all x > 0,where D > 0 is a constant. Suppose that Eφ(|X|) < ∞, E|X|s < ∞ for some1 ≤ s ≤ 2, ||∑ni=1 Xni||/α(n) → 0 in probability, and there exists J ≥ 2 suchthat∞∑n=1φ(α(n))− φ(α(n− 1))n(nαs(n))J< ∞.Then∞∑n=1φ(α(n))− φ(α(n− 1))nP (||n∑i=1Xni|| > εα(n)) < ∞ for all ε > 0.470 SOO HAK SUNG, MANUEL ORDÓÑEZ CABRERA, AND TIEN-CHUNG HU2. Preliminary lemmasLet B be a real separable Banach space with norm || · ||. Let (Ω,F , P )be a probability space. A random element (or B-valued random element) isdefined to be a F-measurable mapping from Ω to B equipped with the Borelσ-algebra (the σ-algebra generated by the open sets determined by || · ||). Theexpected value of a random element X is defined to be Bochner integral (whenE||X|| < ∞) and is denoted by EX.The following lemma is an iterated form of Hoffmann-Jφrgensen [4] inequal-ity and is due to Jain [10].Lemma 1. If X1, . . . , Xn are independent symmetric random elements, thenfor every integer j ≥ 1 and every t > 0P (||Sn|| > 3jt) ≤ CjP ( max1≤i≤n||Xi|| > t) + Dj(P (||Sn|| > t))2j,where Cj and Dj are positive constants depending only on j, and Sn =∑ni=1 Xi.The following lemma gives us a useful contraction principle and can be foundin Lemma 6.5 of Ledoux and Talagrand [13].Lemma 2. Let {Xi, i ≥ 1} be a sequence of symmetric random elements. Letfurther {ξi, i ≥ 1} and {ζi, i ≥ 1} be real random variables such that ξi =φi(Xi), where φi : B → R is symmetric (even), and similarly for ζi. Then, if|ξi| ≤ |ζi| almost surely for every i, for every t > 0P (||∑iξiXi|| > t) ≤ 2P (||∑iζiXi|| > t).In particular, this inequality applies when ξi = I{Xi∈Ai} ≤ 1 ≡ ζi where thesets Ai are symmetric in B (in particular Ai = {||x|| ≤ ai}).The next lemma is a modification of a result of Kuelbs and Zinn [12] concern-ing the relationship between convergence in probability and mean convergencefor sums of independent bounded random variables. We refer to Lemma 2.1 ofHu el al. [7] for the proof.Lemma 3. Let {Xni, 1 ≤ i ≤ kn, n ≥ 1} be an array of rowwise independentsymmetric random elements. Suppose there exists δ > 0 such that ||Xni|| ≤ δalmost surely for all 1 ≤ i ≤ kn, n ≥ 1. Put Sn =∑kni=1 Xni. If Sn → 0 inprobability, then E||Sn|| → 0 as n →∞.The following inequality is a Banach space analogue of the classical Marcink-iewicz-Zygmund inequality and is due to de Acosta [1]. When p = 2, C2 canbe taken to be 4.Lemma 4. Let {Xi, 1 ≤ i ≤ n} be a sequence of independent random elements.Then for 1 < p ≤ 2, there is a positive constant Cp depending only on p suchON COMPLETE CONVERGENCE FOR ARRAYS 471thatE∣∣||Sn|| − E||Sn||∣∣p ≤ Cpn∑i=1E||Xi||p,where Sn =∑ni=1 Xi.Finally, we need the following lemma. The proof is standard and is omitted.Lemma 5. If X and Y have the same distribution, then for every t > 0E||X − Y ||2I(||X − Y || ≤ t) ≤ 8E||X||2I(||X|| ≤ t2) + 2t2P (||X|| > t2).3. ProofsProof of Theorem 3. Let {Xsni, 1 ≤ i ≤ kn, n ≥ 1} be an array of the sym-metrized version of {Xni, 1 ≤ i ≤ kn, n ≥ 1}, i.e., Xsni = Xni − X∗ni, whereXni and X∗ni are independent and have the same distribution. Let µn be amedian of ||∑kni=1 Xni||. By (iv), µn → 0 as n →∞. Then we have by the weaksymmetrization inequality that for all large nP (||kn∑i=1Xni|| > ε) ≤ P (||kn∑i=1Xni|| − µn > ε2) ≤ 2P (||kn∑i=1Xsni|| >ε2)≤ 2P (||kn∑i=1XsniI(||Xsni|| ≤ 2δ)|| >ε4) + 2P (||kn∑i=1XsniI(||Xsni|| > 2δ)|| >ε4)≤ 2P (||kn∑i=1XsniI(||Xsni|| ≤ 2δ)|| >ε4) + 2kn∑i=1P (||Xsni|| > 2δ)≤ 2P (||kn∑i=1XsniI(||Xsni|| ≤ 2δ)|| >ε4) + 4kn∑i=1P (||Xni|| > δ).By (i), it is enough to prove that∞∑n=1anP (||kn∑i=1XsniI(||Xsni|| ≤ 2δ)|| >ε4) < ∞.By Lemma 2 and (iv), we have thatP (||kn∑i=1XsniI(||Xsni|| ≤ 2δ)|| >ε4)≤ 2P (||kn∑i=1Xsni|| >ε4)472 SOO HAK SUNG, MANUEL ORDÓÑEZ CABRERA, AND TIEN-CHUNG HU≤ 2P (||kn∑i=1Xni||+ ||kn∑i=1X∗ni|| >ε4)≤ 4P (||kn∑i=1Xni|| > ε8) → 0.Noting that ||XsniI(||Xsni|| ≤ 2δ)|| ≤ 2δ, it follows by Lemma 3 thatE||kn∑i=1XsniI(||Xsni|| ≤ 2δ)|| → 0as n →∞. Take an integer j such that 2j ≥ J. Then we have by Lemma 1 thatP (||kn∑i=1XsniI(||Xsni|| ≤ 2δ)|| >ε4)≤ CjP ( max1≤i≤kn||XsniI(||Xsni|| ≤ 2δ)|| >ε4 · 3j )+ DjP(||kn∑i=1XsniI(||Xsni|| ≤ 2δ)|| >ε4 · 3j)2j≤ CjP ( max1≤i≤kn||Xsni|| >ε4 · 3j )+ DjP(||kn∑i=1XsniI(||Xsni|| ≤ 2δ)|| >ε4 · 3j)J≤ 2Cjkn∑i=1P (||Xni|| > ε8 · 3j )+ DjP(||kn∑i=1XsniI(||Xsni|| ≤ 2δ)|| >ε4 · 3j)J.Thus, by (i), it suffices to prove that∞∑n=1anP(||kn∑i=1XsniI(||Xsni|| ≤ 2δ)|| >ε4 · 3j)J< ∞.Since E||∑kni=1 XsniI(||Xsni|| ≤ 2δ)|| → 0, it follows by Lemma 4 and Lemma 5that for all large nP (||kn∑i=1XsniI(||Xsni|| ≤ 2δ)|| >ε4 · 3j )≤ P (∣∣||kn∑i=1XsniI(||Xsni|| ≤ 2δ)|| − E||kn∑i=1XsniI(||Xsni|| ≤ 2δ)||∣∣ > ε8 · 3j )ON COMPLETE CONVERGENCE FOR ARRAYS 473≤ (8 · 3jε)2E∣∣∣∣||kn∑i=1XsniI(||Xsni|| ≤ 2δ)|| − E||kn∑i=1XsniI(||Xsni|| ≤ 2δ)||∣∣∣∣2≤ 4(8 · 3jε)2kn∑i=1E||Xsni||2I(||Xsni|| ≤ 2δ)≤ 4(8 · 3jε)2kn∑i=1{8E||Xni||2I(||Xni|| ≤ δ) + 8δ2P (||Xni|| > δ)}.Noting that( ∑kni=1 P (||Xni|| > δ))J≤ O(1) ∑kni=1 P (||Xni|| > δ) by (iii), thecr-inequality implies that∑∞n=1 anP(||∑kni=1 XsniI(||Xsni|| ≤ 2δ)|| > ε4·3j)J<∞ by (i) and (ii). Thus the proof is complete. ¤Proof of Theorem 4. Let A = {n|∑kni=1 P (||Xni|| > δ) ≤ 1}. Define a sequence{un, n ≥ 1} of positive integers byun ={kn, if n ∈ A,1, if n /∈ A.Define an array {Yni, 1 ≤ i ≤ un, n ≥ 1} of random elements byYni ={Xni, if n ∈ A,0, if n /∈ A.Then {Yni, 1 ≤ i ≤ un, n ≥ 1} satisfies all conditions of Theorem 3 and so wehave that∑n∈AanP (||kn∑i=1Xni|| > ε) =∞∑n=1anP (||un∑i=1Yni|| > ε) < ∞.Observe that∞∑n=1ankn∑i=1P (||Xni|| > δ) =∑n∈Aankn∑i=1P (||Xni|| > δ)+∑n/∈Aankn∑i=1P (||Xni|| > δ)≥∑n∈Aankn∑i=1P (||Xni|| > δ) +∑n/∈Aan.474 SOO HAK SUNG, MANUEL ORDÓÑEZ CABRERA, AND TIEN-CHUNG HUIt follows by (i) that∑n/∈A an < ∞. Thus we obtain that∞∑n=1anP (||kn∑i=1Xni|| > ε) =∑n∈AanP (||kn∑i=1Xni|| > ε)+∑n/∈AanP (||kn∑i=1Xni|| > ε)≤∑n∈AanP (||kn∑i=1Xni|| > ε) +∑n/∈Aan < ∞.¤Proof of Corollary 1. We will apply Theorem 4 with an = (φ(α(n)) −φ(α(n−1)))/n, n ≥ 1 and Xni replaced by Xni/α(n), 1 ≤ i ≤ n, n ≥ 1. We only needto verify that conditions (i) and (ii) of Theorem 4 hold. By the weak meandomination hypothesis, we have that∞∑n=1ann∑i=1P (|| Xniα(n)|| > ε) ≤ C∞∑n=1annP (|X|ε> α(n))= C∞∑i=1P (α(i) <|X|ε≤ α(i + 1))i∑n=1nan≤ CEφ( |X|ε) < ∞,since Eφ(|X|) < ∞ and φ(2x) ≤ Dφ(x). Hence condition (i) holds.To establish condition (ii), note that∞∑n=1an( n∑i=1E|| Xniα(n)||2I(|| Xniα(n)|| ≤ δ))J≤ δJ(2−s)∞∑n=1an( n∑i=1E|| Xniα(n)||s)J≤ δJ(2−s)∞∑n=1an(nCE|X|sαs(n))J,since∑ni=1 E||Xni||s ≤ nCE|X|s by the weak mean domination. Hence con-dition (ii) holds. ¤A sequence {Un, n ≥ 1} of random variables is bounded in probability if forevery ε > 0 there exists a constant C > 0 such that P (|Un| > C) < ε for alln ≥ 1.Remark 1. Let {Un, n ≥ 1} be a bounded in probability. Let {βn, n ≥ 1} be asequence of positive real numbers such that βn → 0 as n →∞. Then βnUn → 0in probability.ON COMPLETE CONVERGENCE FOR ARRAYS 475Proof. Let ε > 0 and δ > 0 be given. Since {Un, n ≥ 1} is bounded in proba-bility, there exists a constant C > 0 such that P (|Un| > C) < ε for all n ≥ 1.Since βn → 0 as n →∞, there exists a positive integer N such that βn < δ/Cif n > N. For n > N, P (|βnUn| > δ) ≤ P (|Un| > C) < ε. Hence the proof iscomplete. ¤Remark 2. Let {γn, n ≥ 1} be a sequence of positive real numbers such thatlimn→∞ α(n)/γn = ∞, and {||∑ni=1 Xni||/γn, n ≥ 1} is bounded in probability.By Remark 1, ||∑ni=1 Xni||/α(n) → 0 in probability. When θ(x) = |x|s forsome s > 0,nαs(n)<1r(rn + θ(γn)θ(α(n)))for any r > 0.It follows that Corollary 1 improves Theorem 3.1 of Tómács [15] when θ(x) =|x|s for some 1 ≤ s ≤ 2.References[1] A. de Acosta, Inequalities for B-valued random vectors with applications to the stronglaw of large numbers, Ann. Probab. 9 (1981), no. 1, 157–161.[2] P. Erdős, On a theorem of Hsu and Robbins, Ann. Math. Statist. 20 (1949), 286–291.[3] A. Gut, Complete convergence, Asymptotic Statistics, In: Proceedings of the FifthPrague Symposium, September 4-9, 1993, Physica-Verlag, Wutzburg (1994), 237–247.[4] J. Hoffmann-Jφrgensen, Sums of independent Banach space valued random variables,Studia Math. 52 (1974), 159–186.[5] P. L. Hsu and H. Robbins, Complete convergence and the law of large numbers, Proc.Nat. Acad. Sci. U. S. A. 33 (1947), 25–31.[6] T.-C. Hu, M. Ordóñez Cabrera, S. H. Sung, and A. Volodin, Complete convergencefor arrays of rowwise independent random variables, Commun. Korean Math. Soc. 18(2003), no. 2, 375–383.[7] T.-C. Hu, A. Rosalsky, D. Szynal, and A. Volodin, On complete convergence for arraysof rowwise independent random elements in Banach spaces, Stochastic Anal. Appl. 17(1999), no. 6, 963–992.[8] T.-C. Hu, D. Szynal, and A. Volodin, A note on complete convergence for arrays, Statist.Probab. Lett. 38 (1998), no. 1, 27–31.[9] T.-C. Hu and A. Volodin, Addendum to “A note on complete convergence for arrays”,Statist. Probab. Lett. 38 (1) (1998), 27–31, Statist. Probab. Lett. 47 (2000), no. 2,209–211.[10] N. C. Jain, Tail probabilities for sums of independent Banach space valued randomvariables, Z. Wahrsch. Verw. Gebiete 33 (1975), no. 3, 155–166.[11] A. Kuczmaszewska, On some conditions for complete convergence for arrays, Statist.Probab. Lett. 66 (2004), no. 4, 399–405.[12] J. Kuelbs and J. Zinn, Some stability results for vector valued random variables, Ann.Probab. 7 (1979), no. 1, 75–84.[13] M. Ledoux and M. Talagrand, Probability in Banach spaces, Springer, Berlin, 1991.[14] S. H. Sung, A. Volodin, and T.-C. Hu, More on complete convergence for arrays, Statist.Probab. Lett. 71 (2005), no. 4, 303–311.[15] T. Tómács, Convergence rates in the law of large numbers for arrays of Banach spacevalued random elements, Statist. Probab. Lett. 72 (2005), no. 1, 59–69.476 SOO HAK SUNG, MANUEL ORDÓÑEZ CABRERA, AND TIEN-CHUNG HUSoo Hak SungDepartment of Applied MathematicsPai Chai UniversityTaejon 302-735, KoreaE-mail address: sungsh@pcu.ac.krManuel Ordóñez CabreraDepartment of Mathematical AnalysisUniversity of SevillaSevilla 41080, SpainE-mail address: cabrera@us.esTien-Chung HuDepartment of MathematicsTsing Hua UniversityHsinchu, Taiwan 30043, Republic of ChinaE-mail address: tchu@math.nthu.edu.tw

On complete convergence for arrays of rowwise independent random elements

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On complete convergence for arrays of rowwise independent random elements

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