A complete convergence theorem for row sums from arrays of rowwise independent random elements in Rademacher type p Banach spaces. II. Issue 1 (2nd January 2021) - Hu, Tien-Chung; Rosalsky, Andrew; Volodin, Andrei; Zhang, Sen
Abstract: In this correspondence, for an array of rowwise independent random elements V n, k, 1 ≤ k ≤ k n, n ≥ 1, k n → ∞ taking values in a real separable Rademacher type p ( 1 ≤ p ≤ 2 ) Banach space and a sequence of positive constants c n, n ≥ 1 , the main result provides conditions for the complete convergence result ∑ n = 1 ∞ c n P ( max 1 ≤ k ≤ k n | | ∑ i = 1 k V n, i | | > ε ) < ∞ for all ε > 0 to hold. The complete convergence does not necessary hold if the Rademacher type p hypothesis is dispensed with. Corollaries of the main result are obtained and illustrative examples are presented.