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

2012/03/19 by Robert Knobloch, Knobloch, Robert
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #60F15 #60J25 #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1203.4212

openalex publication_date 2012/03/19 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

In this paper we prove a strong law of large numbers and its L¹-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 Martínez from L¹-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.

Citations

• Polling systems and multitype branching processes in random environment counted by random characteristics

Related

• Strong Law of Large Numbers for Fragmentation Processes
• Limit theorems for fragmentation processes with immigration
• The largest fragment in self-similar fragmentation processes of positive index
• Tail asymptotics for extinction times of self-similar fragmentations
• Fragmentation processes with an initial mass converging to infinity