Laws of Large Numbers for Uncorrelated Set-Valued Random Variables

TL;DR

This paper extends classical laws of large numbers to uncorrelated set-valued random variables in finite-dimensional spaces, using support functions and Hausdorff metric, generalizing previous results for independent cases.

Contribution

It introduces weak and strong laws of large numbers for uncorrelated set-valued random variables, broadening the scope beyond independent cases.

Findings

Proves weak laws of large numbers for uncorrelated set-valued variables.

Establishes strong laws of large numbers in the same context.

Generalizes existing laws from independent to uncorrelated set-valued variables.

Abstract

As the extension of uncorrelated single-valued random variables, set-valued case is studied in this paper. When the underlying space is of finite dimension, by using the support function, We shall prove the weak and strong laws of large numbers for uncorrelated set-valued random variables in the sense of Hausdorff metric $d_H$. Our results generalize weak and strong laws of large numbers for independent identically distributed or independent set-valued random variables.