Convexification in the limit and strong law of large numbers for closed-valued random sets in Banach spaces

TL;DR

This paper establishes a strong law of large numbers for bounded, closed-valued random sets in Banach spaces using Fisher's convergence, which is stronger than Wijsman's but not comparable to Mosco's convergence.

Contribution

It introduces a strong law of large numbers for random sets in Banach spaces utilizing Fisher's convergence, extending previous results to non-separable spaces.

Findings

Proves a strong law of large numbers for random sets in Banach spaces.

Uses Fisher's convergence, a stronger set convergence notion.

Applicable to non-separable Banach spaces.

Abstract

We prove a strong law of large numbers for random sets with bounded and closed values contained in an arbitrary (not necessarily separable) Banach space. We make use of a notion of convergence of sets introduced by Fisher, which is stronger than Wijsman's convergence but in general not comparable with Mosco's convergence.