Further study of complete convergence for weighted sums of PNQD random variables

Qunying Wu

doi:10.1186/s13660-015-0814-1

Abstract

Based on Wu (J. Appl. Math. 2012:104390, 2012), the complete convergence for weighed sums of pairwise negative quadrant dependent (PNQD) random variables is further studied under weaker weighted condition. Sufficient and necessary conditions of complete convergence for weighted sums of PNQD random variables are obtained. Our results generalize and improve those on complete convergence theorems previously obtained by Baum and Katz (Trans. Am. Math. Soc. 120:108-123, 1965), Wu (J. Appl. Math. 2012:104390, 2012) and Zhang (J. Inequal. Appl. 2014:353, 2014).

Highlights

Introduction and lemmas Random variablesX and Y are said to be negative quadrant dependent (NQD) ifP(X ≤ x, Y ≤ y) ≤ P(X ≤ x)P(Y ≤ y) ( . )for all x, y ∈ R
In many mathematical and mechanical models, a pairwise negative quadrant dependent (PNQD) assumption among the random variables in the models is more reasonable than an independence assumption
It is interesting for us to extend the limit theorems to the case of PNQD series

Summary

Introduction

Introduction and lemmas Random variablesX and Y are said to be negative quadrant dependent (NQD) ifP(X ≤ x, Y ≤ y) ≤ P(X ≤ x)P(Y ≤ y) ( . )for all x, y ∈ R. 1 Introduction and lemmas Random variables X and Y are said to be negative quadrant dependent (NQD) if A sequence of random variables {Xn; n ≥ } is said to be pairwise negative quadrant dependent (PNQD) if every pair of random variables in the sequence is NQD.

Results

Conclusion