On Complete Convergence for Arrays of Rowwise Independent Random Elements

A Note on Complete Convergence for Arrays of Rowwise Independent Banach Space Valued Random Elements

“…We note that Theorem 1 has already been obtained in the special case p = 1 by Sung et al [6]. Next, it appears that in the case of symmetric summands Theorems B and C are slightly stronger results than Theorem 1 because they deal with maximums of partial sums.…”

Complete Convergence for Weighted Sums of Random Elements

“…Lemma 3.4 (Sung et al [12]). Let {X ni , i ≥ 1, n ≥ 1} be an array of rowwise independent random elements.…”

A Complete Convergence Theorem for Row Sums from Arrays of Rowwise Independent Random Elements in Rademacher TypepBanach Spaces

“…This result has been generalized and extended in several directions by a number of authors; for results up to 1999, see the discussions in Hu, Rosalsky, Szynal, and Volodin [3]. More recent work on complete convergence is that of Hu and Volodin [4], Hu, Li, Rosalsky, and Volodin [5], Hu, Ordóñez Cabrera, Sung, and Volodin [6], Kuczmaszewska [7], Sung, Volodin, and Hu [8], Kruglov, Volodin, and Hu [9], Sung and Volodin [10], Hernández, Urmeneta, and Volodin [11], Sung, Ordóñez Cabrera, and Hu [12], Sung, Urmeneta, and Volodin [13], Chen, Hernández, Urmeneta, and Volodin [14], and Hu, Rosalsky, and Wang [15]. Some of these generalizations and extensions concern a Banach space setting: a sequence of Banach space valued random elements is said to converge completely to the 0 element of the Banach space if the corresponding sequence of norms converges completely to 0.…”