Geometry Of The Expected Value Set And The Set-Valued Sample Mean Process

TL;DR

This paper investigates the local geometric behavior of the sample mean process for set-valued random variables, providing insights into fluctuations near the boundary of the expected value set.

Contribution

It offers a detailed geometric analysis of the sample mean process, enhancing understanding beyond existing isometric convergence results.

Findings

Describes boundary fluctuations of the sample mean process

Provides geometric insights into convergence behavior

Enhances understanding of set-valued law of large numbers

Abstract

The law of large numbers extends to random sets by employing Minkowski addition. Above that, a central limit theorem is available for set-valued random variables. The existing results use abstract isometries to describe convergence of the sample mean process towards the limit, the expected value set. These statements do not reveal the local geometry and the relations of the sample mean and the expected value set, so these descriptions are not entirely satisfactory in understanding the limiting behavior of the sample mean process. This paper addresses and describes the fluctuations of the sample average mean on the boundary of the expectation set.