Complete moment convergence of weighted sums for processes under asymptotically almost negatively associated assumptions

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1 Proc. Indian Acad. Sci. Math. Sci.) Vol. 124, No. 2, May 214, pp c Indian Academy of Sciences Complete moment convergence of weighted sums for processes under asymptotically almost negatively associated assumptions JUN AN College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 467, China scottan@sina.com MS received 22 January 213; revised 31 January 213 Abstract. For weighted sums of sequences of asymptotically almost negatively associated AANA) random variables, we study the complete moment convergence by using the Rosenthal type moment inequalities. Our results extend the corresponding ones for sequences of independently identically distributed random variables of Chow [4]. Keywords. Asymptotically almost negatively associated; weighted sums; complete moment convergence. 21 Mathematics Subject Classification. 6F15, 6F Introduction A sequence X n,n 1} of random variables is said to be asymptotically almost negatively associated AANA) if there exists a nonnegative sequence qn) asn such that Cov f X n ), gx n1,x n2,...,x nk )) qn) [ Var fx n )Var gx n1,x n2,...,x nk )) ] 1/2 1.1) for all n, k 1 and for all coordinate-wise nondecreasing continuous functions f and g whenever the variances exist. qn),n 1} is called the mixing coefficients of X n, n 1}. Since this concept was introduced by Chandra and Ghosal [6] in 1996, many authors have studied its limit properties. For example, Chandra and Ghosal [6] derived the Kolmogorov type inequality and strong law of large numbers SLLN). Chandra and Ghosal [7] obtained the almost sure convergence of weighted average. Kim et al. [1] established the Hájek Rényi type inequalities and Marcinkiewicz Zygmund type SLLN. Cai [2] investigated the complete convergence of weighted sums. Yuan and An [16] got the Rosenthal type inequalities, L p convergence, complete convergence and Marcinkiewicz Zygmund type SLLN see Wang et al. [14] who obtained the complete convergence and SLLN). Yang et al. [15] derived complete convergence of moving average process for AANA sequence. Recently, An [1] got the following complete convergence of weighted sums for sequences of AANA random variables. 267

2 268 Jun An Theorem A [1]. Let X n,n 1} be a sequence of identically distributed AANA random variables with mixing coefficients qn),n 1},E X 1 p <,αp > 1,α > 1/2. Suppose that EX 1 for p>1. Leta ni,i 1,n 1} be a sequence of real numbers with n a ni r On),r > αp 1)/α 1/2) if p>2; or r 2 if <p 2. Take r 1/2 k 1 2/r)r/r 1) where k is a positive integer number satisfying 2 k <r 2 k1.if q r n) <, then for any ε>, ) n αp 2 P a ni X i >εn α < 1.2) and n αp 2 P sup kn k α ) a ni X i >ε <. 1.3) Theorem B [1]. Let X n,n 1} be a sequence of mean zero, identically distributed AANA random variables with q 2 n) <. Leta ni,i 1,n 1} be a sequence of real numbers satisfying sup n1 n a 2 ni <.IfE X 1 p <, <p<2, then n 1/p a ni X i completely. 1.4) Chow [4] first investigated the complete moment convergence for sequences of independently identically distributed i.i.d.) random variables and obtained the following result. Theorem C [4]. Let X n,n 1} be a sequence of i.i.d. random variables. If <p< 2,r >1,rp 1, then implies E X 1 rp X 1 log1 X 1 )} < 1.5) } n r 1/p 2 E X i kex 1 εn 1/p for any ε>, where and in the following x x,}. < 1.6) Recently, Li and Zhang [11] extended Theorem C to moving average processes under dependent assumptions, Chen and Wang [3] extended Theorem C to identically distributed ϕ-mixing random variables, Zhou and Lin [17] extended Theorem C to moving average processes of ϕ-mixing random variables and Yang et al. [15] extended Theorem C to moving average processes of AANA sequence. The purpose of this paper is to further study the limit properties of weighted sums for sequences of identically distributed AANA random variables and to obtain complete moment convergence by using the Rosenthal type moment inequality. Our results extend the corresponding one of Chow.

3 2. Main results Complete moment convergence of weighted sums 269 Throughout this paper we use the following notations: I ) denotes the indicator function, C stands for a positive constant, its value may be different at different places, represents the Vinogradov symbol O, : means defined as. Theorem 2.1. Let X n,n 1} be a sequence of identically distributed AANA random variables with mean zero and mixing coefficients qn),n 1}, q >,p > 1,αp > 1,α > 1/2. Leta ni,i 1,n 1} be a sequence of real numbers with n a ni r On). r>αp 1)/α 1/2) if p 2, or r 2 if 1 <p<2. q r n) < where r 1/2 k 1 2/r)r/r 1), k is a positive integer number satisfying 2 k <r 2 k1.if E X 1 p < for q p, or E X 1 p log1 X 1 )< for q p, then for any ε>, and } n αp αq 2 q E a ni X i εn α < 2.1) n αp 2 E sup kn k α a ni X i ε } q <. 2.2) Theorem 2.2. Let X n,n 1} be a sequence of mean zero, identically distributed AANA random variables with q 2 n) < and let a ni,i 1,n 1} be a sequence of real numbers satisfying sup n n1 ani 2 <, 1 p<2,q >. IfE X 1 p < for q p, or E X 1 p log1 X 1 )<for q p, then for any ε>, n q/p E a ni X i 1/p}q εn < 2.3) and n q/p E sup a ni X i ε kn } q <. 2.4) The following lemmas are useful for the proofs of our main results. Lemma 2.1 [16].LetX n,n 1} be a sequence of AANA random variables with mixing coefficients qn),n 1}. Letf 1,f 2,... be all nondecreasing or all nonincreasing) functions, then f n X n ), n 1} is still a sequence of AANA random variables with mixing coefficients qn),n 1}. Lemma 2.2 [1,16].LetX n,n 1} be a sequence of AANA random variables with mean zero and mixing coefficients qn),n 1}. Then there exists a positive constant C p depending only on p such that E p n 1 ) p X i C p E X i p q 2 2/p i) X i p 2.5)

4 27 Jun An for all n 1 and 1 <p 2, and such that E p X i C p 1 C ) p 1 q p i) n 1 E X i p EX 2 i ) p/2 2.6) for all n 1 and 2 k < p 2 k1 where integer number k 1, p 1/2 k 1 2/p)p/p 1). In particular, if q 2 n) <, then E p X i C p E X i p 2.7) for all n 1 and 1 <p 2, and if q 1/p 1) n) <, then E p X i C p E X i p EX 2 i ) p/2 2.8) for all n 1 and 3 2 k <p 4 2 k 1 where integer number k 1. Remark 2.1. Since 2 k <p 2 k1 we know 1/2 k 1 2/p 2/p. Byqn) n ) and Hölder inequalities one can easily get 2.6) from 2.2)of[16] see Corollary 2.1 of [1]). Proof of Theorem 2.1. Without loss of generality, we assume a ni foralln 1,i 1. Let Y xi X i I X i x 1/q ) and Y xi X ii X i >x 1/q ). From Lemma 2.1 it is easy to see that for any fixed x>, Y xi,i 1} and a ni Y xi,i 1} are AANA for all n 1. We have n αp αq 2 E a ni X i α}q εn ) n αp αq 2 P a ni X i >εn α x 1/q dx ) n αq n αp αq 2 P a ni X i >εn α x 1/q dx n αp αq 2 n αq P ) a ni X i >εn α x 1/q dx

5 Complete moment convergence of weighted sums 271 ) n αp 2 P a ni X i >εn α n αp αq 2 n αq P ) a ni X i >x 1/q dx : I 1 I ) To prove 2.1) it suffices to prove I 1 < and I 2 <. From Theorem A we have I 1 <. ) I 2 n αp αq 2 P a ni X i >x 1/q dx n αq ) n αp αq 2 P a ni Y n αq xi >x1/q /2 dx ) n αp αq 2 P a ni Y xi >x 1/q /2 dx n αq : I 21 I ) No matter 1 <p<2orp 2, r 2 holds constantly. By C r inequality, [ ) r ] 1/r a ni a ni n r 1 1/r ani) r n. For the first part of 2.1)weget ) I 21 n αp αq 2 x 1/q E a ni Y n αq xi dx n αp αq 2 a ni E X i I X i >x 1/q ) dx n αq x 1/q n αp αq 1 x 1/q E X 1 I X 1 >x 1/q ) dx n αq j1) αq n αp αq 1 x 1/q E X 1 I X 1 >x 1/q ) dx jn j αq n αp αq 1 j αq α 1 E X 1 I X 1 >j α ) jn j αq α 1 E X 1 I X 1 >j α ) j n αp αq )

6 272 Jun An If q p, then I 21 j αp α 1 E X 1 I X 1 >j α ) j αp α 1 E X 1 Il α < X 1 l 1) α ) lj E X 1 Il α < X 1 l 1) α ) l j αp α 1 l αp α E X 1 Il α < X 1 l 1) α )since p>1) E X 1 p Il α < X 1 l 1) α ) E X 1 p <. 2.12) If q p, then j αp α 1 log je X 1 I X 1 >j α ) I 21 j αp α 1 log j E X 1 Il α < X 1 l 1) α ) lj E X 1 Il α < X 1 l 1) α ) l j αp α 1 log j l αp α log le X 1 Il α < X 1 l 1) α )sincep >1) E X 1 p log1 X 1 )I l α < X 1 l 1) α ) E X 1 p log1 X 1 )<. 2.13) Thus I 21 < holds. It is easy to see that a ni EY xi as x ). 2.14) x 1/q So by 2.14)and2.6) of Lemma 2.2 we have ) I 22 n αp αq 2 P a ni Y xi >x 1/q /2 dx n αq ) n αp αq 2 a ni Y xi EY xi ) >x1/q /4 dx n αq P

7 Complete moment convergence of weighted sums 273 r n αp αq 2 x r/q E n αq a ni Y xi EY xi ) dx n αp αq 2 n αq x r/q n αp αq 2 n αq x r/q ) r 1 1 C q r i) E a ni Y xi r dx ) r/2 Ea ni Y xi ) 2 dx : I 221 I ) Since q r n) < and n a r ni On),weget I 221 n αp αq 2 n αq x r/q ani r E Y xi r dx n αp αq 1 x r/q E X 1 r I X 1 x 1/q ) dx n αq j1) αq n αp αq 1 x r/q E X 1 r I X 1 x 1/q ) dx jn j αq n αp αq 1 j αrαq 1 E X 1 r I X 1 j 1) α ) jn j αrαq 1 E X 1 r I X 1 j 1) α ) j n αp αq ) If q p, then I 221 j αp αr 1 E X 1 r I X 1 j 1) α ) j1 j αp αr 1 E X 1 r I l 1) α < X 1 l α ) E X 1 r I l 1) α < X 1 l α ) jl 1 j αp αr 1 l αp αr E X 1 r I l 1) α < X 1 l α )since r>p) E X 1 p I l 1) α < X 1 l α ) E X 1 p <. 2.17)

8 274 Jun An If q p, then I 221 j αp αr 1 log je X 1 r I X 1 j 1) α ) j1 j αp αr 1 log j E X 1 r I l 1) α < X 1 l α ) E X 1 r I l 1) α < X 1 l α ) jl 1 j αp αr 1 log1 j) l αp αr log1 l)e X 1 r I l 1) α < X 1 l α ) E X 1 p log1 X 1 )I l 1) α < X 1 l α ) E X 1 p log1 X 1 )<. 2.18) By C r inequality and r 2, ) r/2 n r/2 1 ani r nr/2. ani 2 The second part of 2.15) can be dominated by r/2 I 222 n αp αq 2 ani 2 EX2 1 I X 1 x )) 1/q dx n αq x r/q n αp αqr/2 2 We consider the following two situations. n αq x r/q EX 2 1 I X 1 x 1/q )) r/2dx. 2.19) i) If p 2, then EX1 2I X 1 x 1/q ) EX1 2 <.Sincer>αp 1)/α 1/2),so for q r, I 222 n αp αqr/2 2 x r/q dx n αq n αp αrr/2 2 <, 2.2) and for q r, I 222 n αp αrr/2 2 log n<. 2.21)

9 Complete moment convergence of weighted sums 275 ii) If 1 <p<2, then r 2. I 222 n αp αq 1 x 2/q EX 2 n αq 1 I X 1 x 1/q ) dx n αp αq 1 jn j1) αq j αq x 2/q EX 2 1 I X 1 x 1/q ) dx n αp αq 1 j αq 2α 1 EX1 2 I X 1 j 1) α ). 2.22) jn Similar to 2.16), 2.17) and2.18) wehavei 222 <. SoI 22 < and I 2 <. The first part of Theorem 2.1 is proved. As for the second part of Theorem 2.1, one can get ) n αp 2 P sup a ni X i >ε x 1/q dx kn k α 2 j 1 n αp 2 P sup n2 j 1 kn k α P sup k2 j 1 k α 2 jαp 1) P sup 2 jαp 1) P 2 jαp 1) P lj 2 lαp 1) P a ni X i >ε x 1/q ) ) a ni X i >ε x 1/q dx k2 j 1 k α sup lj P 2 j 1 dx n2 j 1 2 jαp 2) ) a ni X i >ε x 1/q dx 2 l 1 k<2 l k α 2 l 1 k<2 l k α 2 lαp αq 1) P 2 l 1 k<2 l k α 2 l 1 k<2 l ) a ni X i >ε x 1/q dx ) a ni X i >ε x 1/q dx ) a ni X i >ε x 1/q dx 2 l 1 k<2 l 2 lαp αq 1) P 1k<2 l l 2 jαp 1) ) a ni X i >ε x 1/q )2 l 1)α dx ) a ni X i >2 l 1)α εy 1/q dy letting y 2 l 1)αq x) ) a ni X i > 2 l 1)α ε y 1/q dy

10 276 Jun An 2 l1 1 n αp αq 2 n2 l ) P a ni X i > 2 l1)α 2 2α ε y 1/q dy 1k<2 l 2 l1 1 ) n αp αq 2 P a ni X i >ε n α y 1/q dy n2 l 1k<n ) letting ε 2 2α ε ) n αp αq 2 P a ni X i >ε n α y 1/q dy 1k<n } n αp αq 2 q E a ni X i ε n α <. 2.23) The proof of Theorem 2.1 is completed. Proof of Theorem 2.2. We agree on Y xi and a ni such as the proof of Theorem 2.1. Similarly for the proof of 2.1), we have } n q/p q E a ni X i εn 1/p ) n q/p P a ni X i >εn 1/p x 1/q dx ) n q/p n q/p P a ni X i >εn 1/p dx ) n q/p P a ni X i >x 1/q dx n q/p ) P a ni X i >εn 1/p ) n q/p P a ni X i >x 1/q dx n q/p : I 1 I ) By Theorem B, we know I 1 <.Fromsup n1 n a 2 ni < and C r inequality we get n a ni n 1/2.SinceEX 1, it is easy to see that x 1/q a ni EY xi x 1/q a ni EY xi as x ). 2.25)

11 Complete moment convergence of weighted sums 277 So the second part of 2.24) can be dominated by ) I 2 n q/p a ni Y xi EY ni ) >x 1/q /4 dx n q/p P n q/p n q/p P ) a ni Y xi EY xi ) >x1/q /4 dx ) 2 n q/p x 2/q E a ni Y n q/p xi Y ni ) dx 2 n q/p x 2/q E a ni Y xi Y ni )) dx n q/p n q/p n q/p x 2/q n q/p n q/p x 2/q ani 2 EX2 1 I X 1 >x 1/q ) dx ani 2 EX2 1 I X 1 x 1/q ) dx using 2.7)) n q/p x 2/q EX 2 n q/p 1 I X 1 >x 1/q ) dx n q/p x 2/q EX 2 n q/p 1 I X 1 x 1/q ) dx n q/p jn j1) q/p j q/p x 2/q EX 2 1 I X 1 >x 1/p ) dx n q/p jn j1) q/p j q/p x 2/q EX 2 1 I X 1 x 1/p ) dx n q/p j q/p 2/p 1 EX1 2 I X 1 >j 1/p ) jn n q/p j q/p 2/p 1 EX1 2 I X 1 j 1) 1/p ) jn j q/p 2/p 1 EX1 2 I X 1 >j 1/p ) j n q/p j j q/p 2/p 1 EX1 2 I X 1 j 1) 1/p ) n q/p. 2.26)

12 278 Jun An We consider the following two situations. i) If q p, then I 2 j 2/p EX1 2 I X 1 >j 1/p ) j 2/p EX1 2 I X 1 j 1) 1/p ) j 2/p EX1 2 Il1/p < X 1 l 1) 1/p ) jl j1 j 2/p EX1 2 I l 1)1/p < X 1 l 1/p ) l EX1 2 Il1/p < X 1 l 1) 1/p ) j 2/p EX1 2 I l 1)1/p < X 1 l 1/p ) jl 1 l 2/p1 EX1 2 Il1/p < X 1 l 1) 1/p ) l 2/p1 EX1 2 Il1/p < X 1 l 1) 1/p ) E X 1 p Il 1/p < X 1 l 1) 1/p ) j 2/p E X 1 p <. 2.27) ii) If q p, then I 2 j 2/p log j EX1 2 Il1/p < X 1 l 1) 1/p ) lj j1 j 2/p log j EX1 2 I l 1)1/p < X 1 l 1/p ) l EX1 2 Il1/p < X 1 l 1) 1/p ) j 2/p log j EX1 2 Il1/p < X 1 l 1) 1/p ) j 2/p log1 j) jl 1 l 2/q1 log1 l)ex1 2 Il1/p < X 1 l 1) 1/p )

13 Complete moment convergence of weighted sums 279 E X 1 p log1 X 1 )I l 1/p < X 1 l 1) 1/p ) E X 1 p log1 X 1 )<. 2.28) As for 2.4), we can prove it similar to 2.22). We omit its details. The proof of Theorem 2.2 is completed. Acknowledgments The author is very grateful to the anonymous referee for helpful suggestions on the original manuscript. This work is supported by the Projects of Science and Technology Research of Chongqing City Education Committee. References [1] An J, Complete convergence of weighted sums for sequences of AANA radom variables submitted) [2] Cai G H, Complete convergence for weighted sums of sequences of AANA random variables, Glasgow Math. J. 171) 24) [3] Chen P Y and Wang D C, Complete moment convergence for sequence of identically distributed ϕ-mixing random variables, Acta Math. Sin. Engl. Ser. 264) 21) [4] Chow Y S, On the rate of moment complete convergence of sample sums and extremes, Bull. Inst. Math. Acad. Sin ) [5] Chow Y S and Teicher H, Probability theory: independent, interchangeability, Martingales, third edition 1997) New York: Springer) [6] Chandra T K and Ghosal S, Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund for dependent variables, Acta Math. Hung. 714) 1996a) [7] Chandra T K and Ghosal S, The strong law of large numbers for weighted averages under dependence assumptions, J. Theor. Probab. 93) 1996b) [8] Ghosal S and Chandra T K, Complete convergence of martingale arrays, J. Theor. Probab ) [9] Gut A, Probability theory: a graduate course 25) New York: Springer) [1] Kim T S, Ko M H and Lee I H, On the strong laws for asymptotically almost negatively associated random variables, Rocky Mt. J. Math. 343) 24) [11] Li Y X and Zhang L X, Complete moment convergence of moving average under dependent assumptions, Stat. Probab. Lett. 7 24) [12] Li D L, Rao B, Jiang T F and Wang X C, Complete convergence and almost sure convergence of weighted sums of random variables, J. Theor. Probab ) [13] Petrov V V, Limit theorems of probability: sequences of independent random variables 1995) Oxford: Clarendon Press) [14] Wang X J, Hu S H and Yang W Z, Complete convergence for arrays of rowwise asymptotically almost negatively associated random variables, Discret. Dyn. Nat. Soc., Article ID ) 11 page. doi:1.1155/212/ [15] Yang W Z, Wang X J, Ling N X and Hu S H, On complete convergence of moving average process for AANA sequence, Discret. Dyn. Nat. Soc., Article ID ) 24 pages. doi:1.1155/212/ [16] Yuan D M and An J, Rosenthal type inqualities for asymptotically almost negatively associated random variables and applications, Sci. China Ser. A 529) 29) [17] Zhou X C and Lin J G, Complete q-moment convergence of moving average process under ϕ-mixing assumption, J. Math. Res. Expo. 314) 211)