A Remark on Complete Convergence for Arrays of Rowwise Negatively Associated Random Variables

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1 Proceedings of The 3rd Sino-International Symposium October, 2006 on Probability, Statistics, and Quantitative Management ICAQM/CDMS Taipei, Taiwan, ROC June 10, 2006 pp A Remark on Complete Convergence for Arrays of Rowwise Negatively Associated Random Variables Tien-Chung Hu National Tsing Hua University Andrei Volodin University of Regina ABSTRACT We obtain complete convergence result for arrays of rowwise negatively associated random variables, which extend and generalize the results of Hu et al. 1998, Hu et al. 2003, and Sung et al As applications, some well-known results on independent random variables can be easily extended to the case of negatively associated random variables. Keywords Negatively associated random variables; Array of rowwise negatively associated random variables; Complete convergence. 1. Introduction The concept of complete convergence of a sequence of random variables was introduced by Hsu and Robbins 1947 as follows. A sequence { U n, n 1} of random variables converges completely to the constant θ if n= 1 P{ U n θ > ε} < for all ε > 0. The first results concerning the complete convergence were due to Hsu and Robbins 1947 and Erdös Since then there were many authors who devoted the study to complete convergence for sums and weighted sums of independent random variables. Hu et al announced the general result on complete convergence for arrays of rowwise independent random variables, presented as Theorem A below. Received May 2006, revised August Tien-Chung Hu is a Professor in the Department of Mathematics at the National Tsing Hua University, Hsinchu 300, Taiwan, ROC; tchu@math.nthu.edu.tw. Andrei Volodin is an Associate Professor in the Department of Mathematics and Statistics at the University of Regina, Regina, Saskatchewan, Canada S4S 0A2; andrei@math.uregina.ca. Jointly Published by: The International Chinese Association of Quantitative Management and Chung-Hwa Data Mining Society, Taipei, Taiwan, Republic of China.

2 10 Proceedings of The 3rd Sino-International Symposium on Probability, Statistics, and Quantitative Management 2006 In the following we let f nk ; 1 k ; n 1g be an array of random variables defined on a probability space.; F; P/; f ; n 1g be a sequence of positive integers such that lim D1, and f ; n 1g be a sequence of positive constants. For an event A 2 F we n!1 denote IfAg the indicator function. We should note that all the results of this paper remain true in the case D1for some/all n 1, provided the series P 1 nk converges almost surely. Certainly, we should consider sup instead of max in the case of infinite sums. Theorem A Let f ni ; 1 i ; n 1g be array of rowwise independent random variables. Suppose that for every ">0and some ı>0: i P 1 c P kn n id1 Pfj nij >"g < 1, ii there exists j 2 such that 1 id1 E 2 ni Ifj nij ıg!j < 1; iii P id1 E niifj ni jıg!0 as n!1: Then 1 P j nk j >" < 1 for all ">0: The paper Hu et al. [6] unifies and extends the ideas of previously obtained results on complete convergence. In the main results of Hu et al. [6], no assumptions are made concerning the existence of expected values or absolute moments of the random variables. The proof of Hu et al. [6] is mistakenly based on the fact that the assumptions of their theorem imply convergence in probability of the corresponding partial sums. Counterexamples to this proof were presented in Hu and Volodin [7] and Hu et al. [5]. We would like to stress that the both examples are the counterexamples to the proof of Theorem A, but not to the result. At the same time, they mentioned that the problem whether the Theorem A is true for any positive constants f ; n 1g has remained open. Many attempts to solve this problem led to weaker variants of this theorem. Hu et al. [5] gave a first partial solution to this question. Next partial solution was given by Kuczmaszewska [10] and the question was solved completely in Sung et al. [18], where the result is proved as stated in Hu et al. [6]. The approach of Sung et al. [18] is different from those of Hu et al. [6] in the sense that it does not use the symmetrization procedure. The next paper which, to our best knowledge, presents the most general results on complete convergence for arrays of rowwise independent random variables is Kruglov et al. [9]. As it is shown in corollaries of that paper, the main results of Hu et al. [6], Hu et al. [5], Kuczmaszewska [10], and Sung et al. [18] can be derived easily from the first theorem of Kruglov et al. [9].

3 A Remark on Complete Convergence for Arrays of Rowwise T.-C. Hu and A. Volodin 11 The proofs of the main results of Kruglov et al. [9], as well as of Hu et al. [6], Hu et al. [5], Kuczmaszewska [10], and Sung et al. [18], make use of the well-known Hoffmann-Jørgensen s inequality cf. Hoffmann-Jørgensen [3]. Hoffmann-Jørgensen s maximal inequality is a powerful tool which has now become a standard technique in proving limit theorem for independent random variables. The main purpose of this investigation is to extend Theorem A for the case of arrays of rowwise negatively associated random variables. But for associated random variables, especially for negative associated random variables, it is still an open question whether Hoffmann- Jørgensen s maximal inequality is true or not. In order to extend Theorem A to the case of negatively associated random variables, we need to find another way. In the paper, we use the exponential inequality for negatively associated random variables, which is established by Shao [15], cf. Lemma below. Recall that a finite family of random variables f i ; 1 i ng is said to be negatively associated abbreviated to NA in the following if for any disjoint subsets A and B of f1; 2; ; ng and any real coordinate-wise nondecreasing functions f on R A and g on R B, Cov f. i ; i 2 A/; g. j ; j 2 B/ 0 whenever the covariance exists. An infinite family of random variables f i ; i 1g is NA if every finite subfamily is NA. This concept was introduced by Joag-Dev and Proschan [8]. They also pointed out and proved in their paper that a number of well-known multivariate distributions possess the NA property. NA random variables have wide applications in reliability theory and multivariate statistical analysis. Recently Su et al. [17] showed that NA structure plays an important role in risk management. Because of these reasons the notions of NA random variables have received more and more attention in recent years. A great number of papers for NA random variables are now in literature. We refer to Joag-Dev and Proschan [8] for fundamental properties, Newman [14] for the central limit theorem, Matula [13] for the three series theorem, Shao and Su [16] for the law of the iterated logarithm, Shao [15] for moment inequalities, Liu et al. [12] for the Hàjek-Rènyi inequality and Barbour et al. [1] for the Poison approximation. Shao [15] showed the following important exponent inequality of Kolmogorov s type. Lemma Let f i ; 1 i ng be a sequence of NA mean zero random variables with Ej i j 2 < 1 for every 1 i n and D P n id1 E i 2. Then for all ">0; a > 0 k " 2 P max j i j" 2P max j i j > a C 4 exp 1kn 1in 4.a" C / id1 1 C 23 ln 1 C a" :

4 12 Proceedings of The 3rd Sino-International Symposium on Probability, Statistics, and Quantitative Management 2006 Note also that if f i ; i 1g is a sequence of NA random variables, then the truncation in the usual way, for example the sequence f i Ifj i j <ıg; i 1g, where ı>0, is not necessary NA any more. We should use so-called monotone truncation cf. the definition of random variables Y nk in the proof of Theorem B in order to preserve this property. 2. Main Results, Corollaries, and Remarks Now we state the main results. The proofs will be detailed in next section. Theorem B Let f nk ; 1 k ; n 1g be an array of rowwise NA random variables such that conditions i and ii are satisfied. Then 1 m P Œ nk E. nk Ifj nk jıg/ ˇ ˇ >" < 1 for any ">0. max The following two corollaries of Theorem B are immediate. Corollary 1 Let f nk ; 1 k ; n 1g be an array of rowwise NA random variables such that conditions i, ii, and iii are satisfied. Then 1 P ˇˇˇˇ >" < 1 ˇ nkˇˇˇˇˇ for any ">0. Corollary 2. Let f nk ; 1 k ; n 1g be an array of rowwise NA random variables such that conditions i, ii, and iii 0 max 1mkn j P m id1 E niifj ni jıgj! 0 as n!1 are satisfied. Then 1 m P max >" < 1 ˇ nkˇˇˇˇˇ for any ">0. Similar argument as in Remark 2 of Hu et al. [6], we have Theorem C Let f ni ; 1 i ; n 1g be an array of rowwise NA random variables with E nk D 0 for all n 1 and 1 k. Let.x/ be a real function such that for some ı>0: sup x>ı x.x/ < 1 and Suppose that for all ">0, conditions i and sup 0xı x 2.x/ < 1:

5 A Remark on Complete Convergence for Arrays of Rowwise T.-C. Hu and A. Volodin 13 ii 0 there exist j 2 such that 1 E.j nk j/!j < 1 and E.j nk j/! 0 as n!1 are satisfied, then for any ">0. 1 P max m >" < 1 ˇ nkˇˇˇˇˇ Remark 1. Suppose lim inf n!1 > 0, then condition i in Corollary 2 is also necessary. In fact, 1 m P max >" < 1 ˇ nkˇˇˇˇˇ implies that 1 P max j ni j >" < 1; for all ">0; 1i hence we have P fmax 1ikn j ni j >"g!0 as n!1. By Lemma 2 of Liang [11], we have for sufficiently large n, id1 for some C > 0, therefore condition i holds. Pfj ni j >"gcp max j ni j >" 1i Remark 2. By Theorems B and C we have that Corollaries 1 and 2 of Hu et al. [6] are also true if we assume negative association instead of independence and replace weighted sums by the maxima of weighted sums. 3. Proofs In the following, C always stands for a positive constant which may differ from one place to another. P Proof of Theorem B. The conclusion of the theorem is obvious if 1 < 1. Let fy nk ; 1 k ; n 1g be a monotone truncation of f nk ; 1 k ; n 1g, that is Y nk D nk Ifj nk jıgcıif nk >ıg ıif nk < ıg where ı>0and 1 k ; n 1. Note that by Property 6 of Joag-Dev and Proschan [8] applied twice we can conclude that fy nk ; 1 k ; n 1g is an array of rowwise NA random variables.

6 14 Proceedings of The 3rd Sino-International Symposium on Probability, Statistics, and Quantitative Management 2006 For n 1 and 1 m let T m A D D m m m Y nk ; S m D nk ; S 0 m D nk Ifj nk jıg; \ f nk D nk Ifj nk jıgg ; A c D [ f nk nk Ifj nk jıgg : Note that for any n 1 D Pf max js m ES 0 m j >"g 1mk n P f max js m ES 0 m j >"g\ac C P Pfj nk j >ıgcpf max js 0 m ES0 m j >"g Pfj nk j >ıgcpf max jt m ET m j >"=2g CP f max j.s 0 m T m/ E.S 0 m T m/j >"=2g Pfj nk j >ıgcpf max jt m ET m j >"=2g m CP f max ıj ŒIf nk >ıg If nk < ıg E.If nk >ıg If nk < ıg/ j >"=2g Pfj nk j >ıgcpf max jt m ET m j >"=2g f max js 0 m ES0 m j >"g\a m CCE max j ŒIf nk >ıg If nk < ıg E.If nk >ıg If nk < ıg/ j by Markov s inequality CCE CCE Pfj nk j >ıgcpf max jt m ET m j >"=2g max m ŒIf nk >ıgcif nk < ıgcpf nk >ıgcpf nk < ıg Pfj nk j >ıgcpf max jt m ET m j >"=2g ŒIf nk >ıgcif nk < ıgcpfj nk j >ıg

7 A Remark on Complete Convergence for Arrays of Rowwise T.-C. Hu and A. Volodin 15 Pf max jt m ET m j >"=2gC.1 C 2C / By i it is enough to prove that for all ">0 1 Pfj nk j >ıg: Pf max jt m ET m j >"g < 1: Let D P Var.Y nk/. For any ">0 and a > 0 set Since N 1 DfnW > a"g; N 2 D n W Pfj nk j >ıg > minf1; a"=.2ı 2 /; a=.4ı/g ; N 3 D n W N 4 D N.N 2 [ N 3 /: E 2 nk Ifj nkj ıg > minfa"=2; a 2 =16g ; D.EY 2 nk.ey nk/ 2 / EY 2 nk D ı 2 Pfj nk j >ıgc E 2 nk Ifj nkj ıg; we have N 1 N 2 [ N 3. Note that n2n 2 [N 3 Pf max jt m ET m j >"g minf1=2; a"=.4ı 2 /; a=.8ı/g 1 1 C minfa"=4; a 2 =32g j 1 n2n 2 [N 3 Pfj nk j >ıg E 2 nk Ifj nkj ıg Hence it is sufficient to prove that m P max j.y nk EY nk /j >" < 1: By Lemma we have that P max m.y nk EY nk / ˇ ˇ >"!j < 1:

8 16 Proceedings of The 3rd Sino-International Symposium on Probability, Statistics, and Quantitative Management 2006 Note that for any n 2 N 4 2P max jy nk EY nk j > a 1k " 2 C4 exp 1 C 23 4.a" C / ln 1 C a" max jey nk j max EjY nk j 1k 1k ı max 1k.ıPfj nk j >ıgcej nk jifj nk jıg/ Pfj nk j >ıgc E 2 nk Ifj nkj ıg!1=2 ı minf1; a"=.2ı 2 /; a=.4ı/gc minfa"=2; a 2 =16g 1=2 since n N 2 and n N 3 / a=4 C a=4 D a=2: This implies that for any n 2 N 4 ; max 1kkn jey nk ja=2and P max jy nk EY nk j > a 1k 1 P max jy nk j > a=2 1k 1 Therefore it is sufficient to prove that " 2 exp 4.a" C B n2n n / 4 P fj nk j > minfı; a=2gg < 1 by i. 1 C 23 ln 1 C a" < 1: When n 2 N 4 ; a" and P Pfj nkj >ıg1. Let a D "=.12j /. " exp 1 2 C 23 4.a" C B n2n n / ln 1 C a" 4 exp 1 "2 C 23 8a" ln 1 C a" B n2n n 4 expf 3 2 j g Bn C a" exp j ln D C j Bn C B n2n n C a" 4 Bn a" j D C. / j

9 A Remark on Complete Convergence for Arrays of Rowwise T.-C. Hu and A. Volodin 17 C " C C 8 < ı 2 : ı2j 8 < : ı2j Pfj nk j >ıgc E 2 nk Ifj nkj ıg " kn Pfj nk j >ıg #j #j " 9 #j kn = C E 2 nk Ifj nkj ıg ; " 9 #J kn = Pfj nk j >ıgc E 2 nk Ifj nkj ıg ; since C 1 8 < : ı2j Pfj nk j >ıg1/ Pfj nk j >ıgc " kn 9 #j = E 2 nk Ifj nkj ıg ; < 1 by the assumptions. The proof is completed. Proof of Theorem C. In view of Corollary 2 it is enough to show that conditions ii and iii 0 are satisfied. For ii note that E 2 nk Ifj nkj ıg sup 0xı For iii 0 since E nk D 0 it follows that x 2.x/ E.j nk j/: max j sup x>ı x.x/ m E nk Ifj nk jıgj D max j E.j nk j/! 0 as n!1: m E nk Ifj nk j >ıgj References [1] Barbour, A. D., Holst, L., and Janson, S Poisson approximation, Oxford University Press. [2] Erdös, P On a theorem of Hsu and Robbins, Ann. Math. Statist., 20, [3] Hoffmann-Jørgensen, J Sums of independent Banach space valued random variables, Studia Math., 52, [4] Hsu, P. L. and Robbins, H Complete convergence and the law of large numbers, Proc. Nat. Acad. Sci., 33,

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