Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2012, Article ID 730962, 13 pages doi:10.1155/2012/730962 Research Article Complete Moment Convergence of Weighted Sums for Arrays of Rowwise ϕ-Mixing Random Variables Ming Le Guo School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241003, China Correspondence should be addressed to Ming Le Guo, mleguo@163.com Received 5 June 2012; Accepted 20 August 2012 Academic Editor: Mowaffaq Hajja Copyright q 2012 Ming Le Guo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The complete moment convergence of weighted sums for arrays of rowwise ϕ-mixing random variables is investigated. By using moment inequality and truncation method, the sufficient conditions for complete moment convergence of weighted sums for arrays of rowwise ϕ-mixing random variables are obtained. The results of Ahmed et al. 2002 are complemented. As an application, the complete moment convergence of moving average processes based on a ϕ-mixing random sequence is obtained, which improves the result of Kim et al. 2008. 1. Introduction Hsu and Robbins 1 introduced the concept of complete convergence of {Xn }. A sequence {Xn , n 1, 2, . . .} is said to converge completely to a constant C if ∞ P |Xn − C| > < ∞, ∀ > 0. 1.1 n1 Moreover, they proved that the sequence of arithmetic means of independent identically distributed i.i.d. random variables converge completely to the expected value if the variance of the summands is finite. The converse theorem was proved by Erdös 2. This result has been generalized and extended in several directions, see Baum and Katz 3, Chow 4, Gut 5, Taylor et al. 6, and Cai and Xu 7. In particular, Ahmed et al. 8 obtained the following result in Banach space. 2 International Journal of Mathematics and Mathematical Sciences Theorem A. Let {Xni ; i ≥ 1, n ≥ 1} be an array of rowwise independent random elements in a separable real Banach space B, · . Let P Xni > x ≤ CP |X| > x for some random variable X, constant C and all n, i and x > 0. Suppose that {ani , i ≥ 1, n ≥ 1} is an array of constants such that sup|ani | O n−r , for some r > 0, i≥1 ∞ |ani | Onα , 1.2 for some α ∈ 0, r. i1 Let β be such that α β / − 1 and fix δ > 0 such that 1 α/r < δ ≤ 2. Denote s max1 α β ∞ β 1/r, δ. If E|X|s < ∞ and Sn ∞ i1 ani Xni → 0 in probability, then n1 n P Sn > < ∞ for all > 0. Chow 4 established the following refinement which is a complete moment convergence result for sums of i.i.d. random variables. Theorem B. Let EX1 0, 1 ≤ p < 2 and r ≥ p. Suppose that E|X1 |r |X1 | log1 |X1 | < ∞. Then ∞ n1 r/p−2−1/p n n 1/p E Xi − n < ∞, i1 1.3 ∀ > 0. The main purpose of this paper is to discuss again the above results for arrays of rowwise ϕ-mixing random variables. The author takes the inspiration in 8 and discusses the complete moment convergence of weighted sums for arrays of rowwise ϕ-mixing random variables by applying truncation methods. The results of Ahmed et al. 8 are extended to ϕmixing case. As an application, the corresponding results of moving average processes based on a ϕ-mixing random sequence are obtained, which extend and improve the result of Kim and Ko 9. For the proof of the main results, we need to restate a few definitions and lemmas for easy reference. Throughout this paper, C will represent positive constants, the value of which may change from one place to another. The symbol IA denotes the indicator function of A; x indicates the maximum integer not larger than x. For a finite set B, the symbol B denotes the number of elements in the set B. Definition 1.1. A sequence of random variables {Xi , 1 ≤ i ≤ n} is said to be a sequence of ϕ-mixing random variables, if , P > 0 −→ 0, ϕm sup |P B | A − P B| ; A ∈ k1 , B ∈ ∞ A km k≥1 as m −→ ∞, 1.4 where kj σ{Xi ; j ≤ i ≤ k}, 1 ≤ j ≤ k ≤ ∞. Definition 1.2. A sequence {Xn , n ≥ 1} of random variables is said to be stochastically dominated by a random variable X write {Xi } ≺ X if there exists a constant C, such that P {|Xn | > x} ≤ CP {|X| > x} for all x ≥ 0 and n ≥ 1. International Journal of Mathematics and Mathematical Sciences 3 The following lemma is a well-known result. Lemma 1.3. Let the sequence {Xn , n ≥ 1} of random variables be stochastically dominated by a random variable X. Then for any p > 0, x > 0 E|Xn |p I|Xn | ≤ x ≤ C E|X|p I|X| ≤ x xp P {|X| > x} , 1.5 E|Xn |p I|Xn | > x ≤ CE|X|p I|X| > x. 1.6 Definition 1.4. A real-valued function lx, positive and measurable on A, ∞ for some A > 0, is said to be slowly varying if limx → ∞ lxλ/lx 1 for each λ > 0. By the properties of slowly varying function, we can easily prove the following lemma. Here we omit the details of the proof. Lemma 1.5. Let lx > 0 be a slowly varying function as x → ∞, then there exists C (depends only on r) such that i Ckr1 lk ≤ ii Ckr1 lk ≤ k n1 ∞ nk nr ln ≤ Ckr1 lk for any r > −1 and positive integer k, nr ln ≤ Ckr1 lk for any r < −1 and positive integer k. The following lemma will play an important role in the proof of our main results. The proof is due to Shao 10. Lemma 1.6. Let {Xi , 1 ≤ i ≤ n} be a sequence of ϕ-mixing random variables with mean zero. Suppose 2 that there exists a sequence {Cn } of positive numbers such that E km ik1 Xi ≤ Cn for any k ≥ 0, n ≥ 1, m ≤ n. Then for any q ≥ 2, there exists C Cq, ϕ· such that q kj q/2 q Xi ≤ C Cn E max |Xi | . E max 1≤j≤n k1≤i≤kn ik1 Lemma 1.7. Let {Xi , 1 ≤ i ≤ n} be a sequence of ϕ-mixing random variables with then there exists C such that for any k ≥ 0 and n ≥ 1 E kn ik1 2 Xi ≤C kn ik1 EXi2 . 1.7 ∞ i1 ϕ1/2 i < ∞, 1.8 4 International Journal of Mathematics and Mathematical Sciences Proof. By Lemma 5.4.4 in 11 and Hölder’s inequality, we have E kn 2 kn Xi ik1 ik1 kn ≤ ≤ 1/2 1/2 EXj2 ϕ1/2 j − i EXi2 EXi2 4 k1≤i<j≤kn EXi2 2 kn kn−1 ϕ1/2 j − i EXi2 EXj2 1.9 ik1 ji1 ik1 ≤ EXi Xj k1≤i<j≤kn ik1 kn EXi2 2 14 n 1/2 ϕ i i1 kn EXi2 . ik1 Therefore, 1.8 holds. 2. Main Results Now we state our main results. The proofs will be given in Section 3. Theorem 2.1. Let {Xni , i ≥ 1, n ≥ 1} be an array of rowwise ϕ-mixing random variables with EXni 1/2 m < ∞. Let lx > 0 be a slowing varying function, and {ani , i ≥ 1, n ≥ 0, {Xni } ≺ X and ∞ m1 ϕ 1} be an array of constants such that sup|ani | O n−r , for some r > 0, i≥1 ∞ 2.1 |ani | Onα , for some α ∈ 0, r. i1 a If α β 1 > 0 and there exists some δ > 0 such that α/r 1 < δ ≤ 2, and s max1 α β 1/r, δ, then E|X|s l|X|1/r < ∞ implies k n lnE sup ani Xni − < ∞, k≥1 i1 n1 ∞ β ∀ > 0. 2.2 b If β −1, α > 0, then E|X|1a/r 1 l|X|1/r < ∞ implies k n lnE sup ani Xni − < ∞, k≥1 i1 n1 ∞ −1 ∀ > 0. 2.3 International Journal of Mathematics and Mathematical Sciences 5 Remark 2.2. If α β 1 < 0, then E|X| < ∞ implies that 2.2 holds. In fact, k ∞ ∞ ∞ n lnE sup ani Xni − ≤ nβ ln |ani |E|Xni | nβ ln k≥1 i1 n1 n1 i1 n1 ∞ β ≤C ∞ βα n lnE|X| n1 ∞ 2.4 n ln < ∞. β n1 Remark 2.3. Note that ∞ k ∞ k β ∞> n lnE sup ani Xni − n ln P sup ani Xni − > x dx 0 k≥1 i1 k≥1 i1 n1 n1 ∞ β 2.5 k nβ lnP sup ani Xni > x dx. k≥1 n1 i1 ∞ ∞ 0 Therefore, from 2.5, we obtain that the complete moment convergence implies the complete convergence, that is, under the conditions of Theorem 2.1, result 2.2 implies k β n lnP sup ani Xni > < ∞, k≥1 i1 n1 2.6 k n lnP sup ani Xni > < ∞. k≥1 i1 n1 2.7 ∞ and 2.3 implies ∞ −1 Corollary 2.4. Under the conditions of Theorem 2.1, 1 if α β 1 > 0 and there exists some δ > 0 such that α/r 1 < δ ≤ 2, and s max1 α β 1/r, δ, then E|X|s l|X|1/r < ∞ implies ∞ n lnE ani Xni − < ∞, i1 n1 ∞ β ∀ > 0, 2.8 2 if β −1, α > 0, then E|X|1α/r 1 l|X|1/r < ∞ implies ∞ n lnE ani Xni − < ∞, i1 n1 ∞ −1 ∀ > 0. 2.9 Corollary 2.5. Let {Xni , i ≥ 1, n ≥ 1} be an array of rowwise ϕ-mixing random variables with 1/2 m < ∞. Suppose that lx > 0 is a slowly varying function. EXni 0,{Xni } ≺ X and ∞ m1 ϕ 6 International Journal of Mathematics and Mathematical Sciences 1 Let p > 1 and 1 ≤ t < 2. If E|X|pt l|X|t < ∞, then k 1/t lnE max Xni − n < ∞, 1≤k≤n i1 ∞ p−2−1/t n n1 ∀ > 0. 2.10 ∀ > 0. 2.11 2 Let 1 < t < 2. If E|X|t 1 l|X|t < ∞, then ∞ k 1/t lnE max Xni − n < ∞, 1≤k≤n i1 −1−1/t n n1 Corollary 2.6. Suppose that Xn ∞ i−∞ ain Yi , n ≥ 1, where {ai , −∞ < i < ∞} is a sequence of ∞ real numbers with −∞ |ai | < ∞, and {Yi , −∞ < i < ∞} is a sequence of ϕ-mixing random variables 1/2 m < ∞. Let lx be a slowly varying function. with EYi 0, {Yi } ≺ Y and ∞ m1 ϕ 1 Let 1 ≤ t < 2, r ≥ 1 t/2. If E|Y |r l|Y |t < ∞, then ∞ r/t−2−1/t n n1 n 1/t lnE Xi − n < ∞, i1 ∀ > 0. 2.12 2 Let 1 < t < 2. If E|Y |t 1 l|Y |t < ∞, then ∞ n1 −1−1/t n n 1/t lnE Xi − n < ∞, i1 ∀ > 0. 2.13 Remark 2.7. Corollary 2.6 obtains the result about the complete moment convergence of moving average processes based on a ϕ-mixing random sequence with different distributions. We extend the results of Chen et al. 12 from the complete convergence to the complete moment convergence. The result of Kim and Ko 9 is a special case of Corollary 2.6 1. Moreover, our result covers the case of r t, which was not considered by Kim and Ko. 3. Proofs of the Main Results Proof of Theorem 2.1. Without loss of generality, we can assume sup|ani | ≤ n−r , i≥1 ∞ i1 |ani | ≤ nα . 3.1 International Journal of Mathematics and Mathematical Sciences 7 Let Snk x ki1 ani Xni I|ani Xni | ≤ n−r x for any k ≥ 1, n ≥ 1, and x ≥ 0. First note that E|X|s l|X|1/r < ∞ implies E|X|t < ∞ for any 0 < t < s. Therefore, for x > nr , k −r x n supE|Snk x| x n supE ani Xni I |ani Xni | > n x i1 k≥1 k≥1 −1 r −1 r ≤ ∞ ∞ E|ani Xni |I |ani Xni | > n−r x ≤ E|ani X|I |ani X| > n−r x i1 ≤ EXni 0 i1 3.2 ∞ |ani |E|X|I|X| > x ≤ nα E|X|I|X| > x i1 ≤ xα/r E|X|I|X| > x ≤ E|X|1α/r I|X| > nr −→ 0 as n −→ ∞. Hence, for n large enough we have supk≥1 E|Snk x| < /2n−r x. Then k n lnE sup ani Xni − k≥1 i1 n1 ∞ β ∞ k n ln P sup ani Xni ≥ x dx k≥1 i1 n1 ∞ ∞ β β−r n n1 ≤C ∞ ∞ k −r ln P sup ani Xni ≥ n x dx nr k≥1 i1 β−r n ∞ −r ln P sup|ani Xni | > n x dx nr n1 i ∞ ∞ β−r −r C dx : I1 I2 . n ln P sup|Snk x − ESnk x| ≥ n x 2 nr k≥1 n1 Noting that α β > −1, by Lemma 1.5, Markov inequality, 1.6, and 3.1, we have I1 ≤ C ∞ nβ−r ln nr n1 ≤C ∞ nβ−r ln ∞ nβα ln ≤C n1 i1 nr x−1 ∞ ∞ E|ani Xni |I |ani Xni | > n−r x dx i1 x−1 E|X|I|X| > xdx nr n1 ∞ ∞ P |ani Xni | > n−r x dx nr n1 ≤C ∞ ∞ βα n ln ∞ kr1 kn kr x−1 E|X|I|X| > xdx 3.3 8 International Journal of Mathematics and Mathematical Sciences ≤C ∞ nβα ln n1 ≤C ∞ ∞ k−1 E|X|I|X| > kr ≤ C kn ∞ k−1 E|X|I|X| > kr k nβα ln n1 k1 kβα lkE|X|I|X| > kr ≤ CE|X|11αβ/r l |X|1/r < ∞. k1 3.4 Now we estimate I2 , noting that sup E 1≤m<∞ m ∞ m1 ϕ1/2 m < ∞, by Lemma 1.7, we have −r ani Xni I |ani Xni | ≤ n x − E i1 ≤C ∞ m 2 −r ani Xni I |ani Xni | ≤ n x i1 2 Ea2ni Xni I −r 3.5 |ani Xni | ≤ n x . i1 By Lemma 1.6, Markov inequality, Cr inequality, and 1.5, for any q ≥ 2, we have ≤ Cnrq x−q Esup|Snk x − ESnk x|q P sup|Snk x − ESnk x| ≥ n x 2 k≥1 k≥1 ⎤ ⎡ q/2 ∞ ∞ 2 ≤ Cnrq x−q ⎣ Ea2ni Xni I |ani Xni | ≤ n−r x E|ani Xni |q I |ani Xni | ≤ n−r x ⎦ −r i1 rq −q ≤ Cn x ∞ i1 Ea2ni X 2 I q/2 −r |ani X| ≤ n x Cnrq x−q i1 ∞ 3.6 E|ani X|q I |ani X| ≤ n−r x i1 q/2 ∞ ∞ −r C P |ani X| > n x C P |ani X| > n−r x i1 i1 : J1 J2 J3 J4 . So, I2 ≤ ∞ n1 nβ−r ln ∞ nr J1 J2 J3 J4 dx. ∞ β−r From 3.4, we have ∞ ln nr J4 dx < ∞. n1 n For J1 , we consider the following two cases. 3.7 International Journal of Mathematics and Mathematical Sciences 9 If s > 2, then EX 2 < ∞. Taking q ≥ 2 such that β qα − r/2 < −1, we have ∞ β−r n ∞ ln J1 dx nr n1 ≤C ∞ β−rrq n ∞ ln x ∞ ∞ nr n1 ≤C −q q/2 a2ni 3.8 dx i1 nβ−rrq lnnqα−r/2 nr−q1 ≤ C n1 ∞ nβqα−r/2 ln < ∞. n1 If s ≤ 2, we choose s such that 1 α/r < s < s. Taking q ≥ 2 such that β qr/21 α/r − s < −1, we have ∞ β−r n ∞ ln J1 dx nr n1 ≤C ∞ β−rrq n ∞ ln x nr n1 ≤C ∞ −q ∞ |ani ||ani | i1 nβ−rrq lnnqα/2 n−qr/2s −1 ∞ ∞ x E|ani X| −q n−r x 2−s s q/2 −r |X| I |ani X| ≤ n x q/22−s dx 3.9 dx nr n1 ≤C s −1 nβqr/21α/r−s ln < ∞. n1 ∞ nβ−r ln nr J1 dx < ∞. Now, we estimate J2 . Set Inj {i ≥ 1 | nj 1−r < |ani | ≤ nj−r }, j 1, 2, . . .. Then ∪j≥1 Inj N, where N is the set of positive integers. Note also that for all k ≥ 1, n ≥ 1, So, ∞ n1 nα ≥ ∞ |ani | i1 ≥ ∞ |ani | j1 i∈Inj ∞ ∞ −r −rq ≥ n−r Inj n j 1 Inj j 1 k 1rq−r . j1 3.10 jk Hence, we have ∞ Inj j −rq ≤ Cnαr kr−rq . jk 3.11 10 International Journal of Mathematics and Mathematical Sciences Note that ∞ β−r n ∞ ln J2 dx nr n1 C ∞ β−rrq n ∞ C β−rrq n ln n1 ≤C ∞ C nβ−rrq ln ∞ nβ−r ln ∞ Inj nj ∞ −rq ∞ Inj ∞ nβ−r ln ∞ β−r n ln n1 kr nj ∞ −rq kr−q1−1 r kr−q1−1 E|X|q I |X| ≤ k 1r j 1 ∞ 3.12 Inj j −rq k1j1−1 kr−q1−1 ∞ 2k1−1 E|X|q I ir < |X| ≤ i 1r Inj j −rq i0 j1 k E|X|q I ir < |X| ≤ i 1r i0 j1 ∞ r x−q E|X|q I |X| ≤ x j 1 dx kn kn ∞ k1r kn kn n1 C j1 i∈Inj j1 n1 ≤C E|ani X|q I |ani X| ≤ n−r x dx j1 n1 ∞ ∞ nr n1 ∞ x−q ln r−q1−1 ∞ Inj j −rq j1 kn k1j1 E|X|q I ir < |X| ≤ i 1r i2k1 : J2 J2 . Taking q ≥ 2 large enough such that β α − rq r < −1, for J2 , by Lemma 1.6 and 3.11, we get J2 ≤ C ∞ nβ−r ln n1 C ∞ ∞ kr−q1−1 nαr kr−q1−1 2k1−1 k E|X|q I ir < |X| ≤ i 1r nβα ln n1 2k1−1 E|X|q I ir < |X| ≤ i 1r i0 k1 ∞ i3 ≤CC E|X|q I ir < |X| ≤ i 1r i0 kβα−rqr lk ≤CC 2k1−1 i0 kn k1 ≤C ∞ ∞ i3 ∞ E|X|q I ir < |X| ≤ i 1r kβα−rqr lk ki/2 iβα−rqr1 liE|X|q I ir < |X| ≤ i 1r ≤ C CE|X|1βα1/r l |X|1/r < ∞. 3.13 International Journal of Mathematics and Mathematical Sciences 11 For J2 , we obtain J2 ≤ C ∞ nβ−r ln n1 ≤C ∞ ∞ nβ−r ln C nβ−r ln ∞ ∞ ∞ k−1 ∞ i2k1 ∞ ∞ kβα lk jik1−1 −1 i2k1 kr−q1−1 ∞ E|X|q I ir < |X| ≤ i 1r Inj j −rq nrα ir1−q k−r1−q E|X|q I ir < |X| ≤ i 1r i2k1 3.14 ir1−q E|X|q I ir < |X| ≤ i 1 k r nβα ln n1 i2k1 k1 ≤C kr−q1−1 kn k1 ≤C ∞ j1k1 E|X|q I ir < |X| ≤ i 1r Inj j −rq ∞ j1 kn n1 ∞ kr−q1−1 kn n1 ≤C ∞ ∞ ir1−q E|X|q I ir < |X| ≤ i 1r i2k1 iβα1r−rq E|X|q I ir < |X| ≤ i 1r ≤ CE|X|1βα1/r l |X|1/r < ∞. i4 ∞ ∞ β−r β−r So ∞ ln nr J2 dx < ∞. Finally, we prove ∞ ln nr J3 dx < ∞. In fact, noting n1 n n1 n 1 a/r < s < s and β qr/21 α/r − s < −1, using Markov inequality and 3.1, we get ∞ n1 β−r n ∞ ln J3 dx ≤ C nr ∞ β−r n ln ∞ ∞ nr n1 ≤C ∞ β−r n rs −s n x E|ani X| i1 qrs /2 −rs −1q/2 αq/2 lnn n ∞ n1 q/2 dx ∞ n x−s q/2 dx nr n1 ≤C s nβ−rrq/2αq/2 lnnr−s q/21 ≤ C ∞ nβqr/21α/2−s ln < ∞. n1 3.15 Thus, we complete the proof in a. Next, we prove b. Note that E|X|1α/r < ∞ implies that 3.2 holds. Therefore, from the proof in a, to complete the proof of b, we only need to prove I2 C ∞ n1 −1−r n ∞ −r dx < ∞. ln P sup|Snk x − ESnk x| ≥ n x 2 nr k≥1 3.16 12 International Journal of Mathematics and Mathematical Sciences In fact, noting β −1, α β 1 > 0, α β − r < −1 and E|X|1α/r l|X|1/r < ∞. By taking q 2 in the proof of 3.12, 3.13, and 3.14, we get C ∞ −1r n ∞ ln x−2 ∞ nr n1 Ea2ni X 2 I |ani X| ≤ n−r x dx ≤ C CE|X|1α/r l |X|1/r < ∞. 3.17 i1 Then, by 3.17, we have I2 ≤ C ∞ n−1−r ln nr n1 ≤C ∞ n−1r ln ≤C ∞ n−1r ln ∞ ∞ n−1−r ln n1 ≤C ∞ x−2 n−1r ln x−2 ∞ 2 Ea2ni Xni I |ani Xni | ≤ n−r x dx Ea2ni X 2 I |ani X| ≤ n−r x dx 3.18 i1 ∞ ∞ ∞ nr i1 x−2 nr n1 ∞ i1 nr n1 C n2r x−2 E|Sxn − ESxn |2 dx nr n1 ∞ ∞ P |ani X| > n−r x dx ∞ Ea2ni X 2 I |ani X| ≤ n−r x dx C < ∞. i1 The proof of Theorem 2.1 is completed. Proof of Corollary 2.4. Note that ∞ k ani Xni − ≤ sup ani Xni − . i1 k≥1 i1 3.19 Therefore, 2.8 and 2.9 hold by Theorem 2.1. Proof of Corollary 2.5. By applying Theorem 2.1, taking β p − 2, ani n−1/t for 1 ≤ i ≤ n, and ani 0 for i > n, then we obtain 2.10. Similarly, taking β −1, ani n−1/t for 1 ≤ i ≤ n, and ani 0 for i > n, we obtain 2.11 by Theorem 2.1. Proof of Corollary 2.6. Let Xni Yi and ani n−1/t nj1 aij for all n ≥ 1, −∞ < i < ∞. ∞ 1−1/t . By applying Since −∞ |ai | < ∞, we have supi |ani | On−1/t and ∞ i−∞ |ani | On Corollary 2.4, taking β r/t − 2, r 1/t, α 1 − 1/t, we obtain ∞ n1 r/t−2−1/t n n ∞ ∞ 1/t β lnE Xi − n n lnE a X − < ∞, i1 i−∞ ni ni n1 Therefore, 2.12 and 2.13 hold. ∀ > 0. 3.20 International Journal of Mathematics and Mathematical Sciences 13 Acknowledgment The paper is supported by the National Natural Science Foundation of China no. 11271020 and 11201004, the Key Project of Chinese Ministry of Education no. 211077, the Natural Science Foundation of Education Department of Anhui Province KJ2012ZD01, and the Anhui Provincial Natural Science Foundation no. 10040606Q30 and 1208085MA11. References 1 P. L. Hsu and H. Robbins, “Complete convergence and the law of large numbers,” Proceedings of the National Academy of Sciences of the United States of America, vol. 33, pp. 25–31, 1947. 2 P. Erdös, “On a theorem of hsu and robbins,” Annals of Mathematical Statistics, vol. 20, pp. 286–291, 1949. 3 L. E. Baum and M. 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Volodin, “On the rate of complete convergence for weighted sums of arrays of banach space valued random elements with application to moving average processes,” Statistics & Probability Letters, vol. 58, no. 2, pp. 185–194, 2002. 9 T. S. Kim and M. H. Ko, “Complete moment convergence of moving average processes under dependence assumptions,” Statistics & Probability Letters, vol. 78, no. 7, pp. 839–846, 2008. 10 Q. M. Shao, “A moment inequality and its applications,” Acta Mathematica Sinica, vol. 31, no. 6, pp. 736–747, 1988. 11 W. F. Stout, Almost Sure Convergence, Academic Press, New York, NY, USA, 1974. 12 P. Y. Chen, T. C. Hu, and A. Volodin, “Limiting behaviour of moving average processes under ϕmixing assumption,” Statistics & Probability Letters, vol. 79, no. 1, pp. 105–111, 2009. 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