THÔNG TIN TÀI LIỆU
Statistics and Probability Letters 82 (2012) 1978–1985 Contents lists available at SciVerse ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro Rate of complete convergence for maximums of moving average sums of martingale difference fields in Banach spaces Ta Cong Son a,∗ , Dang Hung Thang a , Le Van Dung b a Faculty of Mathematics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam b Faculty of Mathematics, Danang University of Education, 459 Ton Duc Thang, Lien Chieu, Danang, Viet Nam article info Article history: Received April 2012 Received in revised form 14 June 2012 Accepted 16 June 2012 Available online 25 June 2012 abstract We obtain the rate of complete convergence for maximums of moving average sums of martingale difference fields in p-uniformly smooth Banach spaces, and extend Marcinkiewicz–Zygmund strong laws Our results extend the results of Gut and Stadtmüller (2009), Quang and Huan (2009), Dung and Tien (2010) and some other ones © 2012 Elsevier B.V All rights reserved The authors dedicate the paper to professor Nguyen Duy Tien on the occasion of his seventieth birthday MSC: 60B11 60B12 60F15 60G42 Keywords: Complete convergence Marcinkiewicz–Zygmund strong laws of larger numbers p-uniformly smooth Banach spaces Martingale difference fields Introduction The concept of complete convergence for sums of independent and identically distributed random variables was introduced by Hsu and Robbins (1947) Li et al (1992) and Chen et al (2006) investigated the rate of complete convergence for partial sums for moving average sequences of random variables taking values in Banach spaces of type p Many authors have investigated the Marcinkiewcz–Zygmund strong laws of large numbers for fields {Xn , n ∈ Nd } of random variables For example, Fazekas and Tómács (1998) proved that |n|−1/r Sn → a.s (for some < r < 1) for fields of pairwise independent random variables, and Czerebak-Mrozowicz et al (2002) showed that |n|−1/r (Sn − ESn ) → a.s (for some < r < 2) for fields of pairwise independent random variables Recently, Gut and Stadtmüller (2009) have studied an asymmetric Marcinkiewicz–Zygmund law of large numbers for random fields of i.i.d random variables, Quang and Huan (2009) and Dung and Tien (2010) studied Marcinkiewicz–Zygmund strong laws of large numbers for fields of random variables taking values in p-uniformly smooth Banach spaces ∗ Corresponding author E-mail addresses: congson82@gmail.com (T.C Son), hungthang.dang53@gmail.com (D.H Thang), lvdunght@gmail.com (L.V Dung) 0167-7152/$ – see front matter © 2012 Elsevier B.V All rights reserved doi:10.1016/j.spl.2012.06.014 T.C Son et al / Statistics and Probability Letters 82 (2012) 1978–1985 1979 Let Zd be the integer d-dimensional lattice points, where d is a positive integer Consider a random field of martingale differences {Xn , Fn ; n ∈ Zd } defined on a probability space (Ω , F , P ) taking values in a p-uniformly smooth Banach space E (1 ≤ p ≤ 2) with norm ∥ · ∥ In this paper, we study rate of complete convergence for maximum of moving average sums d of martingale difference fields taking values in p-uniformly smooth Banach spaces n Namely, let {an ; n ∈ Z } be an absolutely summable field of real numbers and set Tk = a X , k ≽ and S = T for n ≽ and we try to find conditions n i∈Zd i i+k k= k to ensure that n ≽1 |n|−1 P { max ∥Sk ∥ > ϵ bn } for every ϵ > 0, 1≼k≼n where {bn , n ≽ 1} is some field of positive constants As a consequence, we establish Marcinkiewicz–Zygmund strong law of large numbers (SLLN) max ∥Sk ∥ → b n ≼ k≼ n a.s as |n| → ∞ (3.3) Throughout this paper, the symbol C will denote a generic constant (0 < C < ∞) which is not necessarily the same one in each appearance Preliminaries and some useful lemmas Throughout this paper, we consider E as a real separable Banach space For a E-valued random variable X and sub σ algebra G of F , the conditional expectation E (X |G) is defined analogously to that in the random variable case and enjoys similar properties (the reader may refer to Scalora, 1961) E is said to be p-uniformly smooth (1 ≤ p ≤ 2) if there exists a finite positive constant C such that for all E-valued martingales {Sn ; n ≥ 1} sup E ∥Sn ∥p ≤ C n ≥1 ∞ E ∥Sn − Sn−1 ∥p n =1 Clearly every real separable Banach space is 1-uniformly smooth and the real line (the same as any Hilbert space) is 2uniformly smooth If a real separable Banach space is p-uniformly smooth for some < p ≤ then it is r-uniformly smooth for all r ∈ [1, p) Let d be a positive integer For m = (m1 , , md ), n = (n1 , , nd ) ∈ Zd , α = (α1 , , αd ) ∈ Nd denote d αi d n d = (2n1 , , 2nd ), |nα | = i=1 ni , = (1, , 1) ∈ N and i=1 (mi < ni ) means that there is at least one of m1 < n1 , m2 < n2 , , md < nd holds We write m ≼ n (or n ≽ m) if mi ≤ ni , ≤ i ≤ d; m ≺ n if m ≼ n and m ̸= n Let {Xn , n ∈ Z d } be a field of E-valued random variables and {Fn , n ∈ Z d } be a field of nondecreasing sub-σ -algebras of F with respect to the partial order ≼ on Z d Assume that Xn is an adapted field with respect to Fn in the sense that Xn is Fn measurable for all n ∈ Z d The notation of martingale difference double array is introduced by Quang and Huan (2009) Naturally, the notation of martingale difference field is defined as follows The field {Xn , Fn , n ∈ Z d } is said to be a martingale difference field if E (Xn |Fn∗ ) = for all n ∈ Z d , where Fn∗ = σ {Fl : d ∨i=1 (li < ni )} To prove the main result we need the following lemmas Lemma 2.1 Let E be a real separable p-uniformly smooth Banach space for some ≤ p ≤ 2, and {Xk , ≼ k ≼ n} be a field of E-random variables We have p ∗ E max (Xi − E (Xi |Fi )) ≤ C E ∥X k ∥p 1≼k≼n 1≼i≼k ≼ k≼ n Proof The proof is completely similar to that of Lemma 1.1 in Quang and Huan (2009) by replacing Skl = ∗ Sn = 1≼i≼k (Xi − E (Xi |Fi )) k l i=1 j =1 Xij by Remark If {Xn ; n ≽ 1} is a E-valued martingale difference field and d = 2, we have Lemma 1.1 in Quang and Huan (2009), and note that {Xn − E (Xn |Fn∗ ); n ≽ 1} is not a E-valued martingale difference field Lemma 2.2 Let < p ≤ 2, α1 , , αd be positive constants, q be the number of integers s such that αs = min{α1 , , αd } and let X be a E-valued random variable Set r = min{α 1, ,α } d 1980 T.C Son et al / Statistics and Probability Letters 82 (2012) 1978–1985 (i) If r < p and E ∥X ∥r (log+ ∥X ∥)q−1 < ∞, then (1) (2) P (∥X ∥ ≥ |nα |) < ∞, n |nα |p P {∥X ∥p ≥ t }dt < ∞ α |p | n n ≽1 (ii) If r > and E ∥X ∥r (log+ ∥X ∥)q−1 < ∞, then ∞ P {∥X ∥ ≥ t }dt < ∞ |nα | |nα | n ≽1 (iii) If r = and E ∥X ∥(log+ ∥X ∥)q < ∞, then ∞ P {∥X ∥ ≥ t }dt < ∞ |nα | |nα | n ≽1 Proof (ii) and (i.2) are proved by Dung and Tien (2010) in Lemma 2.2, and (i.1) is implied from (i.2) by |nα |p |nα |p P {∥X ∥p ≥ t }dt ≥ P (∥X ∥ ≥ |nα |) Now, the proof is similar to that of Lemma 2.2 in Dung and Tien (2010), and we get j ∞ ∞ q −1 jP { j ≤ ∥ X ∥ < j + }( log j ) P {∥ X ∥ ≥ t } dt ≤ C α| α | n k |n | k=1 j =1 n≽1 +C ∞ P {j ≤ ∥X ∥ < j + 1}j(log j)q−1 j =1 ≤C ∞ P {j ≤ ∥X ∥ < j + 1}j(log j)q < ∞ j =1 which implies that (iii) holds Lemma 2.3 Let {bn , n ≽ 1} be a field of positive constants such that bn ≤ bm for all n ≼ m and supn≽1 {Xn , n ≽ 1} be a field of E-valued random variables, and set Sn = b n+1 b2n < ∞ Let 1≼k≼n Xk Then P { max ∥Sk ∥ > ϵ bn } < ∞ for every ϵ > |n| 1≼k≼n n≽1 (2.1) if and only if P n≽1 max ∥Sk ∥ > ϵ b2n ≼ k≼ n < ∞ for every ϵ > (2.2) Moreover, (2.1) implies that the SLLN max ∥Sk ∥ → a.s as |n| → ∞ bn 1≼k≼n holds Proof The proof is similar to that of Theorem 3.1 in Dung and Tien (2010) and so is omitted Let {bn ; n ≽ 1} be an array of positive numbers We define N (x) = Card{n : bn ≤ x}, and suppose that N (x) < ∞, ∀x > Now we define two other functions L(x) and Rp (x) which are little different than those of Su and Tong (2004): L(x) = x N (t ) logd+−1 N (t ) t2 dt and Rp (x) = for x > and p > We have following lemma ∞ x N (t ) logd+−1 N (t ) t p+1 dt , T.C Son et al / Statistics and Probability Letters 82 (2012) 1978–1985 1981 Lemma 2.4 Let {bn ; n ≽ 1} be a field of positive numbers satisfying for each n ≽ 1, bn ≤ bm for all n ≼ m and bn → ∞ as |n| → ∞ Let X be a E-valued random variable (i) If E ∥X ∥L(∥X ∥) < ∞ then ∞ bn n ≽1 P (∥X ∥ > s)ds < ∞ (2.3) bn (ii) If E ∥X ∥p Rp (∥X ∥) < ∞ for some p > then P (∥X ∥ > bn ) < ∞, (2.4) n ≽1 and bn p b n ≽1 n sp−1 P (∥X ∥ > s)ds < ∞ (2.5) Proof First, we prove (i) Suppose that E ∥X ∥L(∥X ∥) < ∞, we have by Lemma of Stadtmüller and Thalmaier (2009) that g (j) = 1∼C j(log j)d−1 1≤n1 , ,nd ≤j as j → ∞ (d − 1)! g (k) = g (k) − g (k − 1) and note that N (x) is non-decreasing Letting s = bn t, we have Denote 1≼n ≼k bn ∞ P (∥X ∥ > s)ds = bn ∞ P ∞ P ≤ ∥X ∥ t g (k)P = N ∞ ∥X ∥ ∥X ∥ ∞ ≤C ∞ j log ≤C EN ∥X ∥ t ∞ ∞ =C N =C j bn dt = t 1≼n ≼k ∥X ∥ dt dP (∥X ∥ ≤ x) dy dP (∥X ∥ ≤ x) = CE ∥X ∥L(∥X ∥) < ∞, for all k ∈ Nd , and then we obtain (2.3) To prove (ii), we have n ≽1 P (∥X ∥ > bn ) ≤ P (N (∥X ∥) ≥ |n|) = ∞ g (k)P (N (∥X ∥) ≥ k) k=1 n ≽1 ≤C ∞ k logd−1 (k)P (k ≤ N (∥X ∥) < k + 1) k=1 ≤ CE (N (∥X ∥) log d−1 N (∥X ∥)) = C · p · E ∥X ∥ p +∞ N (∥X ∥) logd+−1 N (∥X ∥) ∥X ∥ ≤ CE ∥X ∥p Rp (∥X ∥) < ∞ Finally, we easily prove (3.5) by using the method of the proof similar to that of (2.3) t p+1 dt 1982 T.C Son et al / Statistics and Probability Letters 82 (2012) 1978–1985 The field of E-valued random variables {Xn , n ∈ Nd } is said to be stochastically dominated by the E-valued random variable X if, for some < C < ∞, |n| P {∥Xk ∥ ≥ x} ≤ CP {∥X ∥ ≥ x} k≺ n for all n ∈ Nd and x > Main results Let {Xn , Fn ; n ∈ Zd } be a field of E-valued martingale differences Throughout the paper, {an ; n ∈ Zd } is always an absolutely summable field of real numbers such that Tk = i∈Zd Xi+k converges a.s and put Sn = Tk 1≼k≼n The following theorem characterizes the p-uniformly smooth Banach spaces Theorem 3.1 Let ≤ p ≤ 2, and let E be a separable Banach space, then the following two statements are equivalent (i) E is p-uniformly smooth d (ii) For every E-valued martingale difference field {Xn , Fn ; n ∈ Z }, for every absolutely summable field of real numbers d {an ; n ∈ Z } such that Tk = i∈Zd Xi+k converges a.s for all k ≽ and for every field of positive constants {bn , n ≽ 1} such that bn ≤ bm for all n ≼ m and 1< inf 1≼n ≺m b2m ≤ sup b2n n ≽1 b2n+1 b2n < ∞ (3.1) If E ∥Xn ∥p · ϕ(n) < ∞, n∈Zd where ϕ(n) = an−k k≽ p bk , then P {max ∥Sk ∥ > ϵ bn } < ∞ |n| k≼n n≽1 (3.2) for every ϵ > In addition, the SLLN max ∥Sk ∥ → a.s as |n| → ∞ bn 1≼k≼n (3.3) holds Proof We note that Sn = n≽1 ≼ k≼ n Tk = i∈Zd i+1≼j≼i+n Xj By Lemma 2.3, in order to prove [(i) ⇒ (ii)] we show that P { max ∥Sk ∥ > ϵ b2n } < ∞ for every ϵ > 1≼k≼2n Applying the Markov inequality, the Holder inequality and Lemma 2.1 we have that n≽1 P { max ∥Sk ∥ > ϵ b2n } ≤ 1≼k≼2n ≤ ≤ = p |ai | maxn Xj p E ( maxn ∥Sk ∥ ) ≤ p E p p 1≼k≺2 ≼ k≼ ϵ b ϵ b n n 2 n ≽1 n ≽1 i+1≼j≼i+k i∈Zd p p−1 | a | | a | E max X i i j p bp ≼ k≺ n ϵ n n ≽1 i+1≼j≼i+k i∈Zd i∈Zd | | E ∥Xj ∥p (by Lemma 2.1) C p ϵ p b2n i∈Zd n ≽1 i+1≼j≼i+2n p C |ai | E ∥X j ∥ =C | | E ∥ X i + k ∥ p p p p ϵ b2n i∈Zd b n ≽1 k≽1 i∈Zd i+1≼j≼i+2n n:k≼2n 2n p T.C Son et al / Statistics and Probability Letters 82 (2012) 1978–1985 ≤C =C |ai | E ∥ X i +k ∥ p i∈Zd k≼ E ∥Xj ∥p j∈Zd bk |ai | p b i ≼ j −1 j −i p 1983 (by (3.1)) =C E ∥X ∥p · ϕ(j) < ∞ j∈Zd We have (3.2), then (3.3) is implied by (3.2) and Lemma 2.3 Now we prove [(ii) ⇒ (i)] Assume that (ii) holds Let {Yn1 , Gn1 ; n1 ≥ 1} be an arbitrary sequence of martingale difference taking values in E such that ∞ E ∥Yn1 ∥p p n1 n =1 < ∞ For n = (n1 , , nd ) ∈ Z d , set Xn = Yn1 if n1 ≥ 1, n2 = · · · = nd = otherwise Xn = and Fn = σ {Xi ; i ≼ n} Then {Xn , Fn ; n ∈ Z } is the martingale difference field taking values in E Let bn = |n|, = if i ̸= and a0 = so n1 Sn = i=1 Yi and d ∞ E ∥Yn1 ∥p E ∥Xn ∥p ϕ(n) = p n1 =1 n ≽1 n1 < ∞ By (ii), n1 n1 , , nd i1 =1 Xi1 → a.s as |n| → ∞ Taking n2 = · · · = nd = and letting n1 → ∞ we obtain n1 n1 j = Wj → a.s as n1 → ∞ Then by Theorem 2.2 of Hoffmann-Jørgensen and Pisier (1976), E is p-uniformly smooth The next theorem provides an another sufficient condition for (3.2) to hold in terms of functions L(x) and Rp (x) Theorem 3.2 Let {Xn , Fn ; n ∈ Zd } be a martingale difference field taking values in a separable real p-uniformly smooth E Let {bn , n ≽ 1} be a field of positive constants such that bn ≤ bm for all n ≼ m, and bn → +∞ as |n| → +∞ Set N (x) = card{ n; bn ≤ x} for all x > If {Xn ; n ∈ Z d } is stochastically dominated by a E-valued random variable X such that E ∥X ∥p Rp (∥X ∥) < ∞, E ∥X ∥L(∥X ∥) < ∞, then P {max ∥Sk ∥ > ϵ bn } < ∞ |n| k≼n n ≽1 In addition, if supn≽1 b n+1 b2n (3.4) < ∞, then the SLLN max ∥Sk ∥ → a.s as |n| → ∞ b n ≼ k≼ n (3.5) holds Proof For ≼ i ≼ n we set Yni = Xi I{∥Xi ∥>bn } − E (Xi I{∥Xi ∥>bn } |Fi∗ ), Zni = Xi I{∥Xi ∥≤bn } − E (Xi I{∥Xi ∥≤bn } |Fi∗ ) It is clear that Xn = Yni + Zni for all ≼ i ≼ n For an arbitrary ϵ > 0, by using the Chebyshev inequality, Lemmas 2.1 and 2.4 we get max Yni ≥ ϵ bn ≤ |ai |E max Xj ≼ k≼ n 1≼k≼n | n | b ϵ n i+1≼j≼i+k n ≽1 i+1≼j≼i+k i∈Zd i∈Zd ∞ ≤ |ai |E ∥X ∥I{∥X ∥≥bn } ≤ C P (∥X ∥ > s)ds + C P (∥X ∥ > bn ) < ∞ b ϵ b d n ≽1 n n≽1 n bn n≽1 A = P |n| n ≽1 i∈Z 1984 T.C Son et al / Statistics and Probability Letters 82 (2012) 1978–1985 Again using the Chebyshev inequality, the Holder inequality, Lemmas 2.1 and 2.4 we get max Zni ≥ ϵ bn ≼ k≼ n i+1≼j≼i+k i∈Zd p p−1 |ai | |ai |E max ≤ Zni p p ≼ k ≼ n | n | b ϵ n n≽1 i+1≼j≼i+k i∈Zd i∈Zd p−1 p |ai | E ∥Xn ∥ I{∥Xi ∥≤bn } |ai | ≤C |n|bpn ϵ p i∈Zd i+1≼j≼i+n n ≽1 i∈Zd ∞ p sp−1 P (∥X ∥ > s)ds < ∞ ≤C E ∥ X ∥ I ≤ C {∥ X ∥≤ b } p p n B = P |n| n≽1 bn n ≽1 bn n≽1 bn Hence, the conclusion (3.4) holds from P ( max ∥Si ∥ ≥ 2ϵ bn ) ≤ P max ai Zni ≥ ϵ bn < ∞ Yni ≥ ϵ bn + P max 1≼i≼n ≼ k≼ n ≼ k≼ n d d i+1≼j≼i+k i+1≼j≼i+k i∈Z i∈Z The conclusion (3.5) follows from (3.4) and Lemma 2.3 Finally, in the case bn = |nα |, we obtain some new sufficient conditions for (3.2) to hold Theorem 3.3 Let {Xn , Fn ; n ∈ Zd } be a martingale difference field taking values in a separable real p-uniformly smooth E with < p ≤ Let α1 , , αd be positive constants satisfying 1/p < min{α1 , , αd } < 1, let q be the number of integers s such that αs = min{α1 , , αd } If {Xn ; n ∈ Zd } is stochastically dominated by a E-random variable X such that E (∥X ∥r logq−1 ∥X ∥) < ∞ with r = min{α1 , , αd } , then P { max ∥Sk ∥ > ϵ|nα |} < ∞ |n| 1≼k≺n n≽1 (3.6) and the SLLN max ∥Sk ∥ → a.s as |n| → ∞ (3.7) |nα | 1≼k≼n holds Proof For ≼ i ≼ n, we set Yni = Xi I{∥Xi ∥>|nα |} − E (Xi I{∥Xi ∥>|nα |} |Fi∗ ), Zni = Xi I{∥Xi ∥≤|nα |} − E (Xi I{∥Xi ∥≤|nα |} |Fi∗ ) It is clear that Xn = Yni + Zni for all ≼ i ≼ n We have that α α P (max ∥Si ∥ ≥ 2ϵ|n |) ≤ P max Yni ≥ ϵ|n | + P max Zni ≥ ϵ|n | k≼ n k≼ n i≼n d d i+1≼j≼i+k i+1≼j≼i+k α i∈Z i∈Z Then, in order to prove (3.4) it is enough to prove that α max Yni ≥ ϵ|n | < ∞, 1≼k≼n i+1≼j≼i+k i∈Zd α P max B= Zni ≥ ϵ|n | < ∞ ≼ k≼ n |n| d n ≽1 i+1≼j≼i+k A= P |n| n ≽1 i∈Z For A by the same argument as in the proof of Theorem 3.2 and by Lemma 2.2, we have A≤C n ≽1 P {∥X ∥ ≥ |nα |} + C ∞ P {∥X ∥ ≥ t }dt < ∞ |nα | |nα | n ≽1 T.C Son et al / Statistics and Probability Letters 82 (2012) 1978–1985 1985 For B by the same argument as in the proof of Theorem 3.2 and by Lemma 2.2, we have B≤C n ≽1 |nα |p |nα |p P {∥X ∥p ≥ t }dt < ∞ The proof is completed Remark Note that when = if i ̸= and a0 = so Sn = 1≼i≼n Xn then Theorem 3.3 is an extension of Theorem 3.3 in Dung and Tien (2010) and Theorem 2.1 of Gut and Stadtmüller (2009) Theorem 3.4 Let {Xn , Fn ; n ∈ Zd } be a martingale difference field in a separable real p-uniformly smooth E with < p ≤ Let α1 , , αd be positive constants satisfying min{α1 , , αd } = 1, let q be the number of integers s such that αs = = min{α1 , , αd } If {Xn ; n ∈ Zd } is stochastically dominated by a E-random variable X such that E (∥X ∥ logq ∥X ∥) < ∞ Then (3.6) and the SLLN (3.7) hold Proof The proof is similar to that of Theorem 3.3 and using (i) and (iii) of Lemma 2.2 Remark From Theorems 3.3 and 3.5 with = if i ̸= and a0 = 1, d = and α1 = α2 = 1we obtain Theorem 2.4 in Quang and Huan (2009) Acknowledgements The authors would like to express their gratitude to the referee for his/her detailed comments and valuable suggestions which helped them to improve the manuscript The research of the third author (grant no 10103-2012.07) and the second author has been partially supported by Vietnam’s National Foundation for Science and Technology Development (NAFOSTED) References Chen, P., Hu, T.C., Volodin, V., 2006 A note on the rate of complete convergence for maximums of partial sums for moving average processes in Rademachertype p Banach spaces Lobachevskii J Math 21, 45–55 Czerebak-Mrozowicz, E.B., Klesov, O.I., Rychlik, Z., 2002 Marcinkiewicz-type strong laws of large numbers for pairwise independent random fields Probab Math Statist 22 (Fasc 1), 127–139 Dung, L.V., Tien, N.D., 2010 Strong law of large numbers for random fields in martingale type-p Banach spaces Statist Probab Lett 35 (9–10), 756–763 Fazekas, I., Tómács, T., 1998 Strong laws of large numbers for pairwise independent random variables with multidimensional indices Publ Math Debrecen 53 (1–2), 149–161 Gut, A., Stadtmüller, U., 2009 An asymmetric Marcinkiewicz–Zygmund LLN for random fields Statist Probab Lett 79, 1016–1020 Hoffmann-Jørgensen, J., Pisier, G., 1976 The law of large numbers and the central limit theorem in Banach spaces Ann Probab (4), 587–599 Hsu, P.L., Robbins, H., 1947 Complete convergence and the law of large numbers Proc Natl Acad Sci USA 33, 25–31 Li, D., Rao, M.B., Wang, X., 1992 Complete convergence of moving average processes Statist Probab Lett 14, 111–114 Quang, N.V., Huan, N.V., 2009 On the strong laws of large numbers and Lp -convergence for double arrays of random elements in p-uniformly smooth Banach spaces Statist Probab Lett 79, 1891–1899 Scalora, F.S., 1961 Abstract martingale convergence theorems Pacific J Math 11, 347–374 Stadtmüller, U., Thalmaier, M., 2009 Strong laws for delayed sums of random fields Acta Sci Math (Szeged) 75 (3–4), 723–737 Su, C., Tong, T.J., 2004 Almost sure convergence of the general Jamison weighted sum of B -valued random variables Acta Math Sin (Engl Ser.) 20 (1), 181–192 ... defined on a probability space (Ω , F , P ) taking values in a p-uniformly smooth Banach space E (1 ≤ p ≤ 2) with norm ∥ · ∥ In this paper, we study rate of complete convergence for maximum of moving. .. maximum of moving average sums d of martingale difference fields taking values in p-uniformly smooth Banach spaces n Namely, let {an ; n ∈ Z } be an absolutely summable field of real numbers... National Foundation for Science and Technology Development (NAFOSTED) References Chen, P., Hu, T.C., Volodin, V., 2006 A note on the rate of complete convergence for maximums of partial sums for
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