Complete convergence theorems for weighted row sums from arrays of random elements in Rademacher type p and martingale type p Banach spaces. Issue 6 (2nd November 2019) - Hu, Tien-Chung; Rosalsky, Andrew; Volodin, Andrei
Abstract: In this correspondence, complete convergence theorems are established for weighted row sums { ∑ i = 1 j a n, i V n, i, 1 ≤ j ≤ k n, n ≥ 1 } from arrays { V n, i, 1 ≤ i ≤ k n, n ≥ 1, k n → ∞ } of random elements taking values in real separable Rademacher type p   ( 1 ≤ p ≤ 2 ) Banach spaces as well as real separable martingale type p   ( 1 ≤ p ≤ 2 ) Banach spaces. It is assumed that sup { E ‖ V n, i ‖ p : 1 ≤ i ≤ k n, n ≥ 1 } < ∞ . A version of the Rademacher type p complete convergence theorem is also established with random variable weights. Illustrative examples are included.