On the conditions for the complete convergence in mean for double sums of independent random elements in banach spaces

In this paper, we establish the condition for convergence of X∞ m=1 X∞ n=1 1 mn E kSmnk (mn) 1/p q for double arrays of independent random elements in Banach spaces following the type proposed by Li, Qi and Rosalsky. | Trường Đại học Vinh Tạp chí khoa học, Tập 46, Số 2A (2017), tr. 31-42 ON THE CONDITIONS FOR THE COMPLETE CONVERGENCE IN MEAN FOR DOUBLE SUMS OF INDEPENDENT RANDOM ELEMENTS IN BANACH SPACES Vu Thi Ngoc Anh (1) , Nguyen Thi Thuy (2) of Mathematics, Hoa Lu University, Ninh Binh 2 Teacher of Mathematics, Thanh Chuong 3 High School, Nghe An. Received on 19/4/2017, accepted for publication on 20/10/2017 1 Department Abstract: In this paper, we establish the condition for convergence of ∞ X ∞ X 1 kSmn k q E for double arrays of independent random elements mn (mn)1/p m=1 n=1 in Banach spaces following the type proposed by Li, Qi and Rosalsky [6]. 1. Introduction and Preliminaries Throughout this paper, let (B, ) be a real separable Banach space. Li, Qi and Rosalsky [6] extended Theorem in [1], Theorem 5 in [2] and Theorems and in [4] as follows: Let 0 0. Let {Xn , n ≥ 1} be a sequence of independent copies of a B-valued random element X. Set Sn = X1 + X2 + · · · + Xn , n ≥ 1. Then ∞ X 1 kSn k q E t) dt p 0 Based on that idea, we establish the condition for convergence of ∞ ∞ X X 1 kSmn k q E for double arrays of independent random elements in Banach 1/p mn (mn) m=1 n=1 spaces. 1) Email: anhyk86@ (V. T. N. Anh). 31 Vu Thi Ngoc Anh, Nguyen Thi Thuy/ On the conditions for the complete convergence in mean. Throughout this paper, the symbol C denotes a generic constant (0 0. Let {Xmn , m ≥ 1, n ≥ 1} be a double array of independent copies of a B-valued random element X. Set Smn = m X n X Xkl , m ≥ 1, n ≥ 1. k=1 l=1 Then ∞ X ∞ X 1 kSmn k q E p (3) The following three lemmas are used to prove Theorem . Lemma . Let {cmn , m ≥ 1, n ≥ 1} be a double array of nonnegative real numbers. Then i) For all k ≥ 1, l ≥ 1, we have k X l X cij ≤ i=1 j=1 ∞ X ∞ X cij . (4) ckl as n → ∞. (5) i=1 j=1 ii) We have n X n X ckl % k=1 l=1 ∞ X ∞ X k=1 l=1 iii) If {amn , m ≥ 1, n ≥ 1} is a double array of positive .