Complete moment convergence for moving average process generated by ρ -mixing random variables

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1 Zhang Journal of Inequalities and Applications (2015) 2015:245 DOI /s RESEARCH Open Access Complete moment convergence for moving average process generated by ρ -mixing random variables Yong Zhang* * Correspondence: zyong2661@jlu.edu.cn College of Mathematics, Jilin University, Changchun, , P.R. China Abstract Let Yi, < i < } be a sequence of ρ -mixing random variables without the assumption of identical distributions, and ai, < i < } be an absolutely summable sequence of real numbers. In this paper, under some suitable conditions, we establish the complete moment convergence for the partial sum of moving average processes n = ai Yi+n, n 1}. These results promote and improve the corresponding results obtained by Li and Zhang (Stat. Probab. Lett. 70: , 2004) from NA to the case of a ρ -mixing setting. Keywords: complete moment convergence; moving average process; ρ -mixing; Marcinkiewicz-Zygmund strong law of large numbers 1 Introduction Let Yi, < i < } be a sequence of random variables and ai, < i < } be an abso lutely summable sequence of real numbers, and for n set n = ai Yi+n. The limit behavior of the moving average process n, n } has been extensively investigated by many authors. For example, Baek et al. [ ] have obtained the convergence of moving average processes, Burton and Dehling [ ] have obtained a large deviation principle, Ibragimov [ ] has established the central limit theorem, Račkauskas and Suquet [ ] have proved the functional central limit theorems for self-normalized partial sums of linear processes, and Chen et al. [ ], Guo [ ], Kim et al. [, ], Ko et al. [ ], Li et al. [ ], Li and Zhang [ ], Qiu et al. [ ], Wang and Hu [ ], Yang and Hu [ ], Zhang [ ], Zhen et al. [ ], Zhou et al. [ ], Zhou and Lin [ ], Shen et al. [ ] have obtained the complete (moment) convergence of moving average process based on a sequence of dependent (or mixing) random variables, respectively. But very few results for moving average process based on a ρ -mixing random variables are known. Firstly, we recall some definitions. For two nonempty disjoint sets S, T of real numbers, we define dist(s, T) = min j k ; j S, k T}. Let σ (S) be the σ -field generated by Yk, k S}, and define σ (T) similarly. Definition. A sequence Yi, < i < } is called ρ -mixing, if ρ (s) = sup ρ (S, T); S, T Z, dist(s, T) s as s, 2015 Zhang. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 ZhangJournal of Inequalities and Applications (2015) 2015:245 Page 2 of 13 where ρ (S, T)=0 supcorr ( f ( i, i S), g( j, j T) ), where the supremum is taken over all coordinatewise increasing real functions f on R S and g on R T. Definition 1.2 AsequenceY i, < i < } is called ρ -mixing if ρ (s)=sup ρ(s, T); S, T Z, dist(s, T) s } 0 ass, where ρ(s, T)=sup corr(f, g) ; f L2 ( σ (S) ), g L2 ( σ (T) )}. Definition 1.3 AsequenceY i, i Z} is called negatively associated (NA) if for every pair of disjoint subsets S, T of Z and any real coordinatewise increasing functions f on R S and g on R T Cov f (Y i, i S), g(y j, j T) } 0. Definition 1.4 AsequenceY i, < i < } of random variables is said to be stochastically dominated by a random variable Y if there exists a constant C such that P Y i > x } P Y > x }, x 0, < i <. Definition 1.5 A real valued function l(x), positive and measurable on [0, ), is said to be slowly varying at infinity if for each λ >0,lim x l(λx) l(x) =1. Li and Zhang [11] obtained the following complete moment convergence of moving average processes under NA assumptions. Theorem A Suppose that n = a iε i+n, n 1}, where a i, < i < } is a sequence of real numbers with a i < and ε i, < i < } is a sequence of identically distributed NA random variables with Eε 1 =0,Eε1 2 <. Let h be a function slowly varying at infinity,1 q <2,r >1+q/2. Then E ε 1 r h( ε 1 q )< implies n + n h(n)e r/q 2 1/q j εn 1/q < for all ε >0. Chen et al. [20] also established the following results for moving average processes under NA assumptions. Theorem B Let q >0,1 p <2,r 1, rp 1.Suppose that n = a iε i+n, n 1}, where a i, < i < } is a sequence of real numbers with a i < and ε i, < i <

3 ZhangJournal of Inequalities and Applications (2015) 2015:245 Page 3 of 13 } is a sequence of identically distributed NA random variables. If Eε 1 =0and E ε 1 rp <, then n r 2 k P max j εn 1/p < for all ε >0.Furthermore if Eε 1 =0and E ε 1 rp < for q < rp, E ε 1 rp log(1 + ε 1 )< for q = rp, E ε 1 q < for q > rp, then ( n r 2 q/p k + ) q E max j εn 1/q < for all ε >0. Recently, Zhou and Lin [18] obtained the following complete moment convergence of moving average processes under ρ-mixing assumptions. Theorem C Let h be a function slowly varying at infinity, p 1, pα >1and α > 1/2. Suppose that n, n 1} is a moving average process based on a sequence Y i, < i < } of identically distributed ρ-mixing random variables. If EY 1 =0and E Y 1 p+δ h( Y 1 1/α )< for some δ >0,then for all ε >0, n pα 2 α k + h(n)e max j ε < and n pα 2 h(n)e sup k n k α k + j ε} <. Obviously, ρ -mixing random variables include NA and ρ -mixing random variables, which have a lot of applications, their limit properties have aroused wide interest recently, and a lot of results have been obtained; we refer to Wang and Lu [21] for a Rosenthal-type moment inequality and weak convergence, Budsaba et al. [22, 23] for complete convergence for moving average process based on a ρ -mixing sequence, Tan et al. [24] forthe almost sure central limit theorem. But there are few results on the complete moment convergence of moving average process based on a ρ -mixing sequence. Therefore, in this paper, we establish some results on the complete moment convergence for maximum partial sums with less restrictions. Throughout the sequel, C represents a positive constant althoughits valuemay change fromone appearancetothe next, IA} denotes the indicator function of the set A. 2 Preliminary lemmas In this section, we list some lemmas which will be useful to prove our main results. Lemma 2.1 (Zhou [17]) If l is slowly varying at infinity, then

4 ZhangJournal of Inequalities and Applications (2015) 2015:245 Page 4 of 13 m ns l(n) m s+1 l(m) for s > 1and positive integer m, (2) n=m ns l(n) m s+1 l(m) for s < 1and positive integer m. Lemma 2.2 (Wang and Lu [21]) For a positive real number q 2, if n, n 1} is a sequence of ρ -mixing random variables, with E i =0,E i q < for every i 1, then for all n 1, there is a positive constant C = C(q, ρ ( )) such that ( j q) n ( n E max 1 j n i E i q + E 2 i ) q 2 }. Lemma 2.3 (Wang et al. [25]) Let n, n 1} be a sequence of random variables which is stochastically dominated by a random variable. Then for any a >0and b >0, E n a I n b } [ E a I b } + b a P ( > b )], E n a I n > b } E a I > b }. 3 Main results and proofs Theorem 3.1 Let l be a function slowly varying at infinity, p 1, α > 1/2, αp >1.Assume that a i, < i < } is an absolutely summable sequence of real numbers. Suppose that n = a iy i+n, n 1} is a moving average process generated by a sequence Y i, < i < } of ρ -mixing random variables which is stochastically dominated by a random variable Y. If EY i =0for 1/2 < α 1, E Y p l( Y 1/α )< for p >1and E Y 1+δ < for p =1and some δ >0,then for any ε >0 p 2 α k + l(n)e max j ε < (3.1) and p 2 l(n)e sup k n k α k + j ε} <. (3.2) Proof Firstly to prove (3.1). Let f (n)=p 2 α l(n) andy = xiy j < x} + Y j I Y j x} + xiy j > x} and Y (2) = Y j Y be the monotone truncations of Y j, < j < } for x >0. Then by the property of ρ -mixing random variables (cf. Property P2 in Wang and Lu [21]), Y EY, < j < } and Y (2) variables. Note that n k = a i we have for x >,ifα >1 x 1 E i+n a i Y x 1, < j < } are two sequences of ρ -mixing random i+n Y j.since a i <, by Lemma 2.3, i+n [ a i E Yj I Y j x } + xp ( Y > x )] x 1 n [ E Y I Y x } + xp ( Y > x )] n 1 α 0, as n.

5 ZhangJournal of Inequalities and Applications (2015) 2015:245 Page 5 of 13 If 1/2 < α 1, note αp >1,thismeansp >1.ByE Y p l( Y 1/α )<and l is slowly varying at infinity, for any 0 < ɛ < p 1/α,wehaveE Y p ɛ <.ThennotingEY i = 0, by Lemma 2.3 we have x 1 E i+n a i Y = x 1 E x 1 a i i+n Y (2) i+n a i E Y j I Y j > x } x 1 ne Y I Y > x } x 1/α 1 E Y I Y > x } E Y 1/α I Y > x } E Y p ɛ I Y > x } 0, as x. Hence for x > large enough, we get x 1 E i+n a i Y < ε/4. Therefore k + f (n)e max j ε k f (n) P max ε j x dx k f (n) P max j εx dx } i+k f (n) P max a i Y (2) εx/2 dx i+k ( + C f (n) P max a i Y EY ) εx/4 dx =: I 1 + I 2. (3.3) Firstly we show I 1 <. Noting Y (2) < Y j I Y j > x}, by Markov s inequality and Lemma 2.3,wehave I 1 f (n) f (n) x 1 nf (n) x 1 E max a i i+k Y (2) i+n a i E Y (2) dx x 1 E Y I Y > x } dx dx

6 ZhangJournal of Inequalities and Applications (2015) 2015:245 Page 6 of 13 (m+1) α = C nf (n) x 1 E Y I Y > x } dx m α nf (n) m 1 E Y I Y > m α} = C m 1 E Y I Y > m α} m p 1 α l(n). If p >1,thenαp 1 α > 1, and, by Lemma 2.1,weobtain I 1 m αp 1 α l(m)e Y I Y > m α} = C m αp 1 α l(m) E Y I k α < Y (k +1) α} k=m = C E Y I k α < Y (k +1) α} k m αp 1 α l(m) k αp α l(k)e Y I k α < Y (k +1) α} E Y p l ( Y 1/α) <. If p =1,noticethatE Y 1+δ < implies E Y 1+δ l( Y 1/α )< for any 0 < δ < δ, thenby Lemma 2.1,weobtain I 1 m 1 E Y I Y > m α} m n 1 l(n) m 1 E Y I Y > m α} m n 1+αδ l(n) m αδ 1 l(m)e Y I Y > m α} E Y 1+δ l ( Y 1/α) E Y 1+δ <. So, we get I 1 <. (3.4) Next we show I 2 <. By Markov s inequality, the Hölder inequality, and Lemma 2.2, we conclude I 2 f (n) f (n) x r E max [ x r ( E ai r 1 r ( Y EY ) r dx a i i+k ) ( a i 1/r i+k ( max Y EY ) )] r dx

7 ZhangJournal of Inequalities and Applications (2015) 2015:245 Page 7 of 13 + C ( ) r 1 ( f (n) x r a i a i E max f (n) x r i+n a i E Y EY r dx f (n) x r a i ( i+n i+k ( Y E ) r/2 Y EY 2 dx EY ) r) dx =: I 21 + I 22, (3.5) where r 2willbespecializedlater. For I 21,ifp >1,taker > max2, p},thenbyc r inequality, Lemma 2.3, and Lemma 2.1,we get I 21 f (n) x r nf (n) nf (n) i+n [ a i E Yj r I Y j x } + x r P ( Y j > x )] dx x r[ E Y r I Y x } + x r P ( Y > x )] dx (m+1) α m α [ x r E Y r I Y x } + P ( Y > x )] dx [ nf (n) m α(1 r) 1 E Y r I Y (m +1) α} + m α 1 P ( Y > m α)] = C [ m α(1 r) 1 E Y r I Y (m +1) α} + m α 1 P ( Y > m α)] m nf (n) m m α(p r) 1 l(m) E Y r I k α < Y (k +1) α} = C + C m αp 1 l(m) EI k α < Y (k +1) α} k=m E Y r I k α < Y (k +1) α} m α(p r) 1 l(m) + C m=k EI k α < Y (k +1) α} k m αp 1 l(m) k α(p r) l(k)e Y p Y r p I k α < Y (k +1) α} + C k αp l(k)e Y p Y p I k α < Y (k +1) α} E Y p l ( Y 1/α) <. (3.6)

8 ZhangJournal of Inequalities and Applications (2015) 2015:245 Page 8 of 13 For I 21,ifp =1,taker > max1+δ,2},where0<δ < δ, then by the same argument as above we have [ I 21 m α(1 r) 1 E Y r I Y (m +1) α} + m α 1 P ( Y > m α)] m nf (n) [ m α(1 r) 1 E Y r I Y (m +1) α} + m α 1 P ( Y > m α)] m n 1+αδ l(n) [ m α(1 r+δ ) 1 l(m)e Y r I Y (m +1) α} + m α(1+δ ) 1 l(m)ei Y > m α}] E Y 1+δ l ( Y 1/α) E Y 1+δ <. (3.7) For I 22,if1 p <2,taker >2,noteαp + r/2 αpr/2 1 = (αp 1)(1 r/2) < 0, by the C r inequality, Lemma 2.3, and Lemma 2.1,weobtain I 22 n r/2 f (n) x r[( E Y 2 I Y x }) r/2 + x r P r/2( Y > x )] dx (m+1) α n r/2 [ f (n) x r ( E Y 2 I Y x }) r/2 + P r/2 ( Y > x )] dx m α n r/2 [ f (n) m α(1 r) 1 ( E Y 2 I Y (m +1) α}) r/2 + m α 1 P r/2( Y > m α)] [ = C m α(1 r) 1 ( E Y 2 I Y (m +1) α}) r/2 + m α 1 P r/2( Y > m α)] m n r/2 f (n) m α(p r)+r/2 2 l(m) ( E Y p Y 2 p I Y (m +1) α}) r/2 + C m αp+r/2 2 l(m) ( E Y p Y p I Y > m α}) r/2 m αp+r/2 αpr/2 2 l(m) ( E Y p) r/2 <. (3.8) For I 22,ifp 2, take r >(αp 1)/(α 1/2) > 2; we have α(p r)+r/2 2 < 1, and therefore one gets [ I 22 m α(1 r) 1 ( E Y 2 I Y (m +1) α}) r/2 + m α 1 P r/2( Y > m α)] m n r/2 f (n) m α(p r)+r/2 2 l(m) ( E Y 2 I Y (m +1) α}) r/2 + C m αp+r/2 2 l(m) ( E Y 2 Y 2 I Y > m α}) r/2

9 ZhangJournal of Inequalities and Applications (2015) 2015:245 Page 9 of 13 m α(p r)+r/2 2 l(m) ( E Y 2) r/2 <. (3.9) Thus, (3.1) can be deduced by combining (3.3)-(3.9). Now, we show (3.2). By Lemma 2.1 and (3.1)wehave p 2 l(n)e sup = p 2 l(n) k n k α 0 k j ε P sup } + k n k α k j > ε + t dt 2 i 1 = p 2 k l(n) P sup n=2 i 1 0 k n k α j > ε + t dt k 2 i 1 P sup 0 k 2 i 1 k α j > ε + t dt p 2 l(n) n=2 i 1 2 i(αp 1) l ( 2 i) k P sup j > ε + t dt 2 i(αp 1) l ( 2 i) l=1 0 P 0 l=i max 0 k 2 i 1 k α P 2 l 1 k<2 l k α 0 max 2 l 1 k<2 l k α k j > ε + t dt k j > ε + t dt l 2 i(αp 1) l ( 2 i) 2 l(αp 1) l ( 2 l) k P max j >(ε + t)2 (l 1)α dt l=1 0 2 l 1 k<2 l 2 l(αp 1 α) l ( 2 l) k P max l=1 0 1 k<2 l j > ε2 (l 1)α + y dy p 2 α k l(n) P max j > ε 2 α + y dy 1 k<n = C p 2 α k + l(n)e max j ε 0 <. 1 k<n Hence the proof of Theorem 3.1 is completed. The next theorem treats the case αp =1. Theorem 3.2 Let l be a function slowly varying at infinity, 1 p <2.Assume that a i θ <, where θ belong to (0, 1) if p =1and θ =1if 1<p <2.Suppose that n = a iy i+n, n 1} is a moving average process generated by a sequence Y i, < i < } of ρ -mixing random variables which is stochastically dominated by a random variable Y.

10 ZhangJournal of Inequalities and Applications (2015) 2015:245 Page 10 of 13 If EY i =0and E Y p l( Y p )<, then for any ε >0 n 1 1/p k + l(n)e max j εn 1/p <. (3.10) Proof Let g(n)=n 1 1/p l(n).similarlytotheproofof(3.3), we have k + g(n)e max j εn 1/p g(n) P max n 1/p + C g(n) P max n 1/p i+k a i Y (2) a i i+k } εx/2 dx ( Y EY ) εx/4 dx =: J 1 + J 2. (3.11) For J 1, by Markov s inequality, the C r inequality, Lemma 2.3, and Lemma 2.1,onegets J 1 g(n) x θ E max n 1/p a i i+k ng(n) x θ E Y θ I Y > x } dx n 1/p = C ng(n) (m+1) 1/p m 1/p Y (2) θ dx x θ E Y θ I Y > x } dx ng(n) m (1 θ)/p 1 E Y θ I Y > m 1/p} = C m (1 θ)/p 1 E Y θ I Y > m 1/p} m ng(n) m θ/p l(m)e Y θ I Y > m 1/p} = C m θ/p l(m) E Y θ I k 1/p < Y <(k +1) 1/p} = C k=m E Y θ I k 1/p < Y <(k +1) 1/p} k m θ/p l(m) k 1 θ/p l(k)e Y θ I k 1/p < Y <(k +1) 1/p} E Y p l ( Y p) <. (3.12)

11 ZhangJournal of Inequalities and Applications (2015) 2015:245 Page 11 of 13 For J 2, similar to the proof of I 2,taker = 2, by Lemma 2.2, Lemma 2.3, and Lemma 2.1,we conclude J 2 g(n) x 2 E max n 1/p ( Y EY ) 2 dx a i i+k ng(n) x 2[ E Y 2 I Y x } + x 2 P ( Y > x )] dx n 1/p = C ng(n) (m+1) 1/p m 1/p x 2[ E Y 2 I Y x } + x 2 P ( Y > x )] dx [ ng(n) m 1 1/p E Y 2 I Y (m +1) 1/p} + m 1/p 1 P ( Y > m 1/p)] = C [ m 1 1/p E Y 2 I Y (m +1) 1/p} + m 1/p 1 P ( Y > m 1/p)] m ng(n) [ m 2/p l(m)e Y 2 I Y (m +1) 1/p} + l(m)p ( Y > m 1/p)] E Y p l ( Y p) <. (3.13) Hence from (3.11)-(3.13), (3.10)holds. For the complete convergence and strong law of large numbers, we have the following corollary from the above theorems immediately. Corollary 3.3 Under the assumptions of Theorem 3.1, for any ε >0we have p 2 k l(n)p max j > ε <. (3.14) Under the assumptions of Theorem 3.2, for any ε >0we have n 1 k l(n)p max j > εn 1/p < ; (3.15) in particular, the assumptions EY i =0and E Y p < imply the following Marcinkiewicz- Zygmund strong law of large numbers: lim n 1 n 1/p n j =0 a.s. (3.16) Remark 3.4 Corollary 3.3 provides complete convergence for the maximum of partial sums, which extends the corresponding results of Budsaba et al. [22, 23] andtheorem1 of Baek et al. [1] with less restrictions. Since ρ -mixing random variables include NA and ρ -mixing random variables, our results also hold for NA and ρ -mixing, and therefore

12 ZhangJournal of Inequalities and Applications (2015) 2015:245 Page 12 of 13 Theorem 3.1 improves upon the above Theorem A from Li and Zhang [11] with less restrictions, and our results also extend and generalize the above Theorem B from Chen et al. [20]withq =1partly. Remark 3.5 Obviously, the assumption that Y i, < i < } is stochastically dominated by a random variable Y is weaker than the assumption of identical distribution of the random variables Y i, < i < }, therefore the above results also hold for identically distributed random variables. Remark 3.6 Let a 0 =1,a i =0,i 0,thenS n = n k = n Y k.hencetheaboveresults hold when k, k 1} is a sequence of ρ -mixing random variables which is stochastically dominated by a random variable Y. Competing interests Theauthordeclarestohavenocompetinginterests. Acknowledgements This work was supported by National Natural Science Foundation of China (Grant No ) and the Science and Technology Development Program of Jilin Province (Grant No JH). Received: 17 April 2015 Accepted: 23 July 2015 References 1. Baek, JI, Kim, TS, Liang, HY: On the convergence of moving average processes under dependent conditions. Aust. N. Z. J. Stat. 45, (2003) 2. Burton, RM, Dehling, H: Large deviations for some weakly dependent random processes. Stat. Probab. Lett. 9, (1990) 3. Ibragimov, IA: Some limit theorem for stationary processes. Theory Probab. Appl. 7, (1962) 4. Račkauskas, A, Suquet, C: Functional central limit theorems for self-normalized partial sums of linear processes. Lith. Math. J. 51(2), (2011) 5. Chen, PY, Hu, TC, Volodin, A: Limiting behaviour of moving average processes under ϕ-mixing assumption. Stat. Probab. Lett. 79, (2009) 6. Guo, ML: On complete moment convergence of weighted sums for arrays of row-wise negatively associated random variables. Stochastics 86(3), (2014) 7. Kim, TS, Ko, MH: Complete moment convergence of moving average processes under dependence assumptions. Stat. Probab. Lett. 78(7), (2008) 8. Kim, TS, Ko, MH, Choi, YK: Complete moment convergence of moving average processes with dependent innovations. J. Korean Math. Soc. 45(2), (2008) 9. Ko, MH, Kim, TS, Ryu, DH: On the complete moment convergence of moving average processes generated by ρ -mixing sequences. Commun. Korean Math. Soc. 23(4), (2008) 10. Li, DL, Rao, MB, Wang, C: Complete convergence of moving average processes. Stat. Probab. Lett. 14, (1992) 11. Li, Y, Zhang, L: Complete moment convergence of moving average processes under dependence assumptions. Stat. Probab. Lett. 70, (2004) 12. Qiu, DH, Liu, D, Chen, PY: Complete moment convergence for maximal partial sums under NOD setup. J. Inequal. Appl. 2015, 58 (2015) 12 pp 13. Wang, J, Hu, SH: Complete convergence and complete moment convergence for martingale difference sequence. Acta Math. Sin. Engl. Ser. 30, (2014) 14. Yang, WZ, Hu, SH: Complete moment convergence of pairwise NQD random variables. Stochastics 87(2), (2015) 15. Zhang, L: Complete convergence of moving average processes under dependence assumptions. Stat. Probab. Lett. 30, (1996) 16. Zhen,, Zhang, LL, Lei, YJ, Chen, ZG: Complete moment convergence for weighted sums of negatively superadditive dependent random variables. J. Inequal. Appl. 2015,117 (2015) 17. Zhou, C: Complete moment convergence of moving average processes under ϕ-mixing assumptions. Stat. Probab. Lett. 80, (2010) 18. Zhou, C, Lin, JG: Complete moment convergence of moving average processes under ρ-mixing assumption. Math. Slovaca 61(6), (2011) 19. Shen, AT, Wang, H, Li, Q, Wang, J: On the rate of complete convergence for weighted sums of arrays of rowwise φ-mixing random variables. Commun. Stat., Theory Methods 43, (2014) 20. Chen, PY, Hu, TC, Volodin, A: Limiting behaviour of moving average processes under negative association assumption. Theory Probab. Math. Stat. 77, (2007) 21. Wang, JF, Lu, FB: Inequalities of maximum of partial sums and weak convergence for a class of weak dependent random variables. Acta Math. Sin. 22, (2006) 22. Budsaba, K, Chen, PY, Volodin, A: Limiting behavior of moving average processes based on a sequence of ρ mixing random variables. Thail. Stat. 5,69-80 (2007)

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