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

Abstract

We obtain complete convergence results for arrays of rowwise independent Banach space valued random elements. Compared with similar results presented in the probabilistic literature our conditions are weaker.

Complete Convergence for Weighted Sums of Random Elements

“…No assumptions are given on the geometry of the underlying Banach space. The result generalizes the main results of Ahmed et al [1], Chen et al [2], and Volodin et al [14]. …”

“…Chen et al [2] improved Theorem 1.4 by proving that the condition The plan of the paper is as follows. In Section 2, we recall well known inequalities and give some elementary results pertaining to the current work.…”

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.…”