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On strong law of large numbers for weakly stationary $\varphi$-mixing set-valued random variable sequences

Luc Tri Tuyen

TL;DR

This work addresses strong laws of large numbers for weakly stationary phi-mixing sequences of set-valued random variables in a Banach space. It introduces a set-valued phi-mixing framework and proves LLNs for the normalized partial sums, yielding almost sure convergence in the Hausdorff metric to the Aumann expectation A and, under additional hypotheses, convergence in Kuratowski-Mosco sense to D = closure(co) A. The approach leverages scalar support functions and existing LLN results for scalar phi-mixing sequences to transfer dependence properties to the set-valued setting. The results extend the LLN theory for random sets and are illustrated by examples demonstrating naturalness and sharpness; future work includes exploring rho-mixing and alternative summability methods.

Abstract

In this paper we extend the notion of $\varphi$-mixing to set-valued random sequences that take values in the family of closed subsets of a Banach space. Several strong laws of large numbers for such $\varphi$-mixing sequences are stated and proved. Illustrative examples show that the hypotheses of the theorems are both natural and sharp.

On strong law of large numbers for weakly stationary $\varphi$-mixing set-valued random variable sequences

TL;DR

This work addresses strong laws of large numbers for weakly stationary phi-mixing sequences of set-valued random variables in a Banach space. It introduces a set-valued phi-mixing framework and proves LLNs for the normalized partial sums, yielding almost sure convergence in the Hausdorff metric to the Aumann expectation A and, under additional hypotheses, convergence in Kuratowski-Mosco sense to D = closure(co) A. The approach leverages scalar support functions and existing LLN results for scalar phi-mixing sequences to transfer dependence properties to the set-valued setting. The results extend the LLN theory for random sets and are illustrated by examples demonstrating naturalness and sharpness; future work includes exploring rho-mixing and alternative summability methods.

Abstract

In this paper we extend the notion of -mixing to set-valued random sequences that take values in the family of closed subsets of a Banach space. Several strong laws of large numbers for such -mixing sequences are stated and proved. Illustrative examples show that the hypotheses of the theorems are both natural and sharp.
Paper Structure (4 sections, 4 theorems, 84 equations)

This paper contains 4 sections, 4 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Main results
  4. Conclusions

Key Result

theorem 1.1

gan2007strong Let $\{x_n\}_{n\ge1}$ be a $\varphi$-mixing sequence with finite second moments and $\sum_{n=1}^{\infty}\varphi^{1/2}(n)<\infty$. If $\{a_n\}_{n\ge1}$ is a non-decreasing sequence of positive numbers with $a_n\to\infty$ and $\sum_{n=1}^{\infty}\!Var(x_n)/a_n^{2}<\infty$, then

Theorems & Definitions (11)

  • theorem 1.1
  • definition 3.1
  • definition 3.2
  • remark 3.1
  • theorem 3.1
  • proof
  • theorem 3.2
  • proof
  • theorem 3.3
  • proof
  • ...and 1 more