Asymptotic properties of the process counted with a random characteristic in the context of fragmentation processes

TL;DR

This paper establishes strong laws of large numbers for self-similar fragmentation processes with random characteristics, extending previous results and applying to energy and empirical mean analyses.

Contribution

It extends Nerman's law of large numbers to fragmentation processes with infinite dislocation measures without discretization.

Findings

Proves strong law of large numbers for fragmentation with random characteristics

Extends limit theorem for fragmentation energy from L^1 to almost sure convergence

Analyzes asymptotic behavior of empirical means in stopped fragmentation processes

Abstract

In this paper we prove a strong law of large numbers and its L^1-convergence counterpart for the process counted with a random characteristic in the context of self-similar fragmentation processes. This result extends a somewhat analogical result by Nerman for general branching processes to fragmentation processes. In addition, we apply the general result of this paper to a specific example that in particular extends a limit theorem, concerning the fragmentation energy, by Bertoin and Mart\'inez from L^1-convergence to almost sure convergence. Our approach treats fragmentation processes with an infinite dislocation measure directly, without using a discretisation method. Moreover, we obtain a result regarding the asymptotic behaviour of the empirical mean associated with some stopped fragmentation process.