The strong law of large numbers for arrays of NA random variables

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1 International Mathematical Forum,, 2006, no. 2, 83-9 The strong law of large numbers for arrays of NA random variables K. J. Lee, H. Y. Seo, S. Y. Kim Division of Mathematics and Informational Statistics, Institute of Basic Natural Science, Wonkwang University Iksan, Chonbuk Republic of Korea D. H. Ryu Department of Computer Science ChungWoon University Republic of Korea Abstract Let {X ni i n, n } be an array of rowwise negatively associated random variables under some suitable conditions. Then it is shown that for some 2 <t, n /t max ki= k n X ni 0 completely as n if and only if E X 2t < and E X ni = 0 and r n max j kn j i= X ni 0 completely as n implies E X k+ r <. AMS Mathematics Subject Classification: Primary 60F05 ; Secondary 62E0, 45E0. Keywords and phrases: Negatively associated random variables, Strong law of large numbers, Complete convergence. Introduction The concept of negatively associated random variables was introduced by Joag- Dev and Proschan [7] although a very special case was first introduced by Lehmann[9]. Many authors derived several important properties about negatively associated NA sequences and also discussed some applications in the area of statistics, probability, reliability and multivariate analysis. Compared

2 84 K. J. Lee, H. Y. Seo, S. Y. Kim, D. H. Ryu to positively associated random variables, the study of NA random variables has received less attention in the literature. Readers may refer to Karlin and Rinott[8], Ebrahimi and Ghosh[3], Block et al.[2], Newman[2], Joag- Dev[6], Joag-Dev and Proschan[7], Matula[] and Roussas[3] among others. Recently, some authors focused on the problem of limiting behavior of partial sums of NA sequences. Su et al.[4] derived some moment inequalities of partial sums and a weak convergence for a strongly stationary NA sequence. Su and Qin[5] studied some limiting results for NA sequences. More recently, Liang and Su[0], and Baek, Kim and Liang[] considered some complete convergence for weighted sums of NA sequences. Let {X nk } be an array of random variables with EX nk = 0 for all n and k and let p<2. Then n /p X nk 0 completely as n. k= and where complete convergence is defined Hsu and Robbins [4] by n= P n /p Xnk >ε < for each ε>0..2 Hu, Móricz and Taylor[5] showed that for an array of i.i.d. random variables {X nk },. holds if and only if E X 2p <. The main purpose of this paper is to extend a similar results above to rowwise NA random variables, since independent and identically random variables are a special case of NA random variables. That is, we investigate the necessary and sufficient condition for max n /t k n k i= X ni 0 completely as n where <t and let {k 2 n} and {r n } be two increasing positive sequences satisfying some conditions, then, we show that r n max j kn j i= X ni 0 completely as n implies E X k+ r < in NA setting. Finally, in order to prove the strong law of large numbers for array of NA random variables, we give an important definition and some lemmas which will be used in obtaining the strong law of large numbers in the next section. Definition.[7]. Random variables X,,X n are said to be negatively associated NA if for any two disjoint nonempty subsets A and A 2 of {,,n} and f and f 2 are any two coordinatewise nondecreasing functions, Cov f X i, i A,f 2 X j,j A 2 0, whenever the covariance is finite. An infinite family of random variables is NA If every finite subfamily is NA.

3 The strong law of large numbers for arrays of NA random variables 85 Lemma.2[0]. Let {X i i } be a sequence of NA random variables and {a ni i n, n } be an array of real numbers. If P max j n a nj X j > ε <δ for δ small enough and n large enough, then j= for sufficient large n. P a nj X j >ε=op max j n a njx j >ε Lemma.3[5]. For any r, E X r < if and only if More precisely, r2 r n= n r P X >n <. n= n r P X >n E X r +r2 r n r P X >n. n= Lemma.4[5]. If r and t>0, then and n /t E X r I X n /t r t r P X >tdt 0 E X I X >n /t =n /t P X >n /t + n /t P X >tdt. 2 Main results Theorem 2.. Let 2 <t and let {X ni i n, n } be an array of rowwise NA random variables such that EX ni = 0 and P X ni >x= OP X >x for all x 0. If E X 2t <, then n /t max X ni 0 completely as n. k n i= Proof. We define that for i n, n and 2 <t Y ni = X ni I X ni n /t +n /t IX ni >n /t n /t IX ni < n /t.

4 86 K. J. Lee, H. Y. Seo, S. Y. Kim, D. H. Ryu To prove Theorem 2., it suffices to show that P max X ni Y ni εn /t < for all ε>0, k n n= i= i= 2. max n EY /t ni 0, k n i= 2.2 P max Y ni εn /t <, for all ε>0. k n n= i= 2.3 The proofs of can be found in the following Lemmas Lemma 2.. If E X 2t <, then 2. holds. Proof. when 2 <t since E X 2t <. P max X ni Y ni εn /t k n n= i= i= n P X ni Y ni n= i= P X ni >n /t n= i= = OnP X >n /t n= O E X 2t <, Lemma 2.2. If E X 2t < and EX ni = 0, then n /t max EY ni 0. k n i= Proof. To prove max n /t k n k i= EY ni 0, it suffices to show that n= max n /t k n k i= EY ni <. Note that by EX ni = 0, we have max n= n /t k n i= n= n /t i= =: I + I 2 say. EY ni E X ni I X ni >n /t + n= n /t n /t P X ni >n /t i=

5 The strong law of large numbers for arrays of NA random variables 87 First, to estimate I, by using Lemma.4, [ ] I = n /t P X n= n /t ni >n /t + P X ni >xdx i= n /t = O np X >n /t n +O P X >xdx =: I n /t say. n /t n= n= Letting x = n /t s and applying Lemma.3, we have I = O np X >n /t +O n P X >n /t sds n= n= OE X 2t + O np s X t >nds n= OE X 2t + OE X 2t s 2t ds = OE X 2t <. As to I 2, we have I 2 = n /t P X n= n /t ni >n /t i= = P X ni >n /t n= i= = O np X >n /t n= OE X 2t <. Lemma 2.3. If E X 2t <, then P max Y ni εn /t < for all k n n= i= ε>0. Proof. From the definition of NA random variables, we know that {Y ni i k, n } is still an array of rowwise NA random variables. Thus, we obtain that n= P max k n O O n= n= i= n E n Y ni εn /t max k n i= E Y ni t i= t Y ni

6 88 K. J. Lee, H. Y. Seo, S. Y. Kim, D. H. Ryu [ O n= n i= =: I 3 + I 4 say. E X ni t I X ni n /t + n= n i= ] P X ni >n /t First, we prove that I 3 <. Let G ni x =P X ni x, then we have I 3 = O E X ni t I X ni n /t n= n i= n /t O x n= i= 0 n /t t dg ni x n /t = O dg ni xds n= i= 0 ns /t = O P ns /t < X ni <n /t ds O n= i= 0 n= OE X 2t <. 0 np X > ns /t ds Also, the proof of I 4 is similar to that of Lemma 2.3. Corollary below is a corresponding result for a sequence of rowwise NA random variables. Corollary. Let 2 <t and let {X i i } be a sequence of NA random variables such that EX i = 0 for all i and P X i >x=op X >x for all x 0. If E X 2t <, then max X n /t i 0 completely as n. k n i= Theorem 2.2. Let <t and let {X 2 ni i n, n } be an array of rowwise NA random variables such thatp X >x=o P X ni >x for all x 0. Assume that max ki=0 n /t k n X ni 0 completely as n, then E X 2t < and EX ni =0. Proof. From the assumptions, for any ε>0, P max X ni εn /t <, 2.4 k n n= i= By Lemma.2, we obtain that P X ni ɛn /t =OP max X ni ɛn /t, k n i= i=

7 The strong law of large numbers for arrays of NA random variables 89 which, together with 2.4 and assumptions, we have np X ɛn /t < n= which is equivalaent to E X 2t <, by Lemma.3. Now, under E X 2t <, we obtain from Theorem 2. that P max X ni EX ni ɛn /t < for any ɛ>0 2.5 k n n= i= 2.4 and 2.5 yield EX ni =0 Theorem 2.3. Let {X ni i k n,n } be an array of rowwise NA random variables with C P X >x C inf n,i P X ni >x C 2 sup n,i P X ni >x C 2 P X >x for all x 0. Assume that {k n } and {r n } are two sequences satisfying r n b n r,k n b 2 n k, for some b,b 2,r,k >0. Let j max r n j k n i= X ni 0 completely as n. If k +<r,then E X k+ r <. Proof. Note that j max r X ni 0 completely as n. n j n i= i.e. n= P max j k n Since max j k n X nj 2 max j X ni εr n <, for all ε> i= j j k n i= n= X ni, 2.6 implies P max X nj n <, 2.7 j n and P max X nj r n j k n 0 as n. 2.8 By 2.7 and 2.8, and using Lemma.2, we obtain that k n i= P X ni >r n =OP max X nj r n, j k n

8 90 K. J. Lee, H. Y. Seo, S. Y. Kim, D. H. Ryu which, together with 2.7, it follows that k n n= i= P X ni >r n <. Thus, using the assumptions of Theorem 2.3, we have k n P X >b n r <, n= which is equivalent to E X k+ r <. Corollary 2. Let {X ni i k n,n } be an array of rowwise identically distributed NA random variables. Assume that {k n } and {r n } are two sequences satisfying r n n r,k n n k, for some r, k > 0 where a n b n means that C a n b n C 2 a n for large enough n. If k +<ror 2 r k +<trfor some 0 <t< and EX 2 ni = 0, then r n max j kn j i= X ni 0 completely as n if and only if E X k+ r <. ACKNOWLEDGEMENTS.This paper was supported by Wonkwang University Research in References [] J.I. Baek, T.S. Kim and H.Y. Liang, On the convergence of moving average processes under dependent conditions, Australian and New Zealand Journal of Statistics, , 3, [2] H.W. Block, T.H.Savits and M.Shaked, Some concepts of negative dependence, The Annals of Probability, 0982, [3] N. Ebrahimi and M.Ghosh, Multivariate negative dependence, Communications in Statistics. A. Theory and Methods, 098, [4] P.L. Hsu, H. Robbins, Complete convergence and the law of large numbers, Proceedings of the National Academy of Sciences of the United States of America, 33947, [5] T.C. Hu, F. Móricz, R.L.Taylor, Strong laws of large numbers for arrays of rowwise independent random variables, Statistics Technical Report 27. University of Georgia,986

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