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)

• Record Type: Journal Article
• Authors: Hu, Tien-Chung; Rosalsky, Andrew; Volodin, Andrei; Zhang, Sen
• Is Part Of: Stochastic analysis and applications. Volume 39:Issue 1(2021)
• Page Start: 177
• Page End: 193
• Publication Date: 2021-01-02
• Subjects: Complete convergence -- array of Banach space valued random elements -- Rademacher type p Banach space -- rowwise independent; Primary 60F15 -- 60B12 -- Secondary 60B11; Stochastic analysis -- Periodicals; 519.2205
• DOI: 10.1080/07362994.2020.1791721
• Languages: English
• ISSNs: 0736-2994

Abstract

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.