A NOTE ON THE COMPLETE MOMENT CONVERGENCE FOR ARRAYS OF B-VALUED RANDOM VARIABLES

Abstract

Abstract. Inthisarticle,wediscussthecompletemomentconvergencefor arrays of B-valued random variables. We obtain some new resultswhichimprovethecorrespondingonesofSungandVolodin[17]. 1. IntroductionLet {Ω,F,P} be a probability space, and let B be a separable real Banachspace with norm ||·||. A random element is defined to be an F-measurablemapping of Ω into B equipped with the Borel σ-algebra (that is, the σ-algebragenerated by the open sets determined by ||·||). The expected value of a B-valued random element X is defined to be the Bochner integral and denotedby EX.Let {X nk ,k ≥ 1,n ≥ 1} be an array of random elements in a real Banachspace. An array of rowwise random elements {X nk ,k ≥ 1,n ≥ 1} is said to bestochastically dominated by a random variable X (write {X nk } ≺ X) if thereexists a constant C > 0 such thatsup n≥1,k≥1 P(||X nk || > x) ≤ CP(|X| > x), ∀x > 0.Now we recall the following concepts of convergence which were introducedby Hsu and Robbins [6] and Chow [4], respectively.Definition 1.1. A sequence of random variables {X