Complete Moment Convergence for Sung’s Type Weighted Sums ofB-Valued Random Elements

Wei Li,Pingyan Chen,Soo Hak Sung

doi:10.1155/2016/1484160

Abstract

Letp≥1/αand1/2<α≤1.Let{X,Xn, n≥1}be a sequence of independent and identically distributedB-valued random elements and let{ani, 1≤i≤n, n≥1}be an array of real numbers satisfying∑i=1naniq=O(n)for someq>p.We give necessary and sufficient conditions for complete moment convergence of the form∑n=1∞n(p-v)α-2E∑i=1naniXi-εnα+v<∞, ∀ε>0, where0<v<p.A strong law of large numbers for weighted sums of independentB-valued random elements is also obtained.

Highlights

Let {Xn, n ≥ 1} be a sequence of random variables and let real numbers
We refer to Bai and Su [14], Baum and Katz [15], Chen [16], Chen et al [17, 18], Chen and Wang [10], Katz [19], Li and Spătaru [11, 20], Qiu et al [21], Rosalsky et al [22], Sung [23], Wang and Su [24], and Wu et al [25]
The purpose of this paper is to extend Theorem A to complete moment convergence for independent and identically distributed random elements taking values in a Banach space B

Summary

Introduction

Let {Xn, n ≥ 1} be a sequence of random variables (or random array of elements) and let real numbers. {X, Xn, 1} be an n ≥ 1} be a array of real sequence of independent and identically distributed numbers satisfying ∑ni=1 |ani|q = O(n) for some q > p. N ≥ random variables and let {ani, 1 ≤ i ≤ n, n ≥ 1} be an array of real numbers with sup n n−1∑

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Results

Conclusion