Convergence Theorems for m -Coordinatewise Negatively Associated Random Vectors in Hilbert Spaces

Lyurong Shi

doi:10.1155/2021/3462317

Abstract

In this study, some new results on convergence properties for m -coordinatewise negatively associated random vectors in Hilbert space are investigated. The weak law of large numbers, strong law of large numbers, complete convergence, and complete moment convergence for linear process of H-valued m -coordinatewise negatively associated random vectors with random coefficients are established. These results improve and generalise some corresponding ones in the literature.

Highlights

Introduction e random variablesX1, X2, . . . , Xn are said to be negatively associated (NA, in short) if, for every pair of disjoint subsets A and B of {1, 2, . . . , n} and any real coordinatewise nondecreasing functions f1 on R|A| and f2 on R|B|, Cov􏼐f1 Xi, i ∈ A􏼁, f2􏼐Xj, j ∈ B􏼑􏼑 ≤ 0, (1)whenever the covariance above exists, where |A| and |B| denote the cardinalities of A and B, respectively
[3] extended the concept of NA random variables to m-NA random variables; Zhang and Wang [4] generalised it to a more broad case, i.e., asymptotically negative association (ANA); Zhang [5] extended it to Rd-valued random vectors; Ko et al [6] introduced the concept of NA random vectors taking values in real separable Hilbert spaces
Let 􏼈Xi, − ∞ < i < ∞􏼉 be a sequence of zero mean H-valued m-coordinatewise negatively associated (CNA) random vectors coordinatewise weakly upper bounded by a random vector X with 􏽐j∈BE|X(j)|p < ∞

Summary

Introduction

Introduction e random variablesX1, X2, . . . , Xn are said to be negatively associated (NA, in short) if, for every pair of disjoint subsets A and B of {1, 2, . . . , n} and any real coordinatewise nondecreasing (or nonincreasing) functions f1 on R|A| and f2 on R|B|, Cov􏼐f1 Xi, i ∈ A􏼁, f2􏼐Xj, j ∈ B􏼑􏼑 ≤ 0, (1)whenever the covariance above exists, where |A| and |B| denote the cardinalities of A and B, respectively. Baum–Katz type complete convergence with 1 ≤ p < 2 and αp > 1, which partially extends eorem A to CNA random vectors in Hilbert space. Inspired of by random variables is Hu et al [3] and Huan et al [18], we introduce the concept of m-CNA random vectors in Hilbert space as follows. Let 􏼈Xi, − ∞ < i < ∞􏼉 be a sequence of m-CNA random vectors.

Results

Conclusion