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

TL;DR

This paper extends the strong law of large numbers to weakly stationary set-valued random sequences with $$-mixing properties in Banach spaces, providing new theoretical results and illustrative examples.

Contribution

It introduces a novel extension of $$-mixing to set-valued sequences and establishes strong laws of large numbers for these sequences.

Findings

Proved strong laws of large numbers for $$-mixing set-valued sequences.

Examples demonstrate the naturalness and sharpness of the hypotheses.

Extended classical results to a broader set-valued and Banach space context.

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.