Functional limit theorems for sums of independent geometric L\'{e}vy processes

TL;DR

This paper establishes functional limit theorems for sums of independent geometric Lévy processes, revealing different limiting behaviors—either Ornstein-Uhlenbeck or stable processes—depending on the growth rate of the sequence involved.

Contribution

It introduces new functional limit theorems for sums of independent geometric Lévy processes, extending previous results and characterizing limits based on growth rates.

Findings

Limit process is Ornstein-Uhlenbeck if growth is slow.

Limit process is an asymmetric stable process for intermediate growth.

Different limit behaviors depend on the growth rate of the sequence.

Abstract

Let $\xi_i$, $i\in \mathbb {N}$, be independent copies of a L\'{e}vy process $\{\xi(t),t\geq0\}$. Motivated by the results obtained previously in the context of the random energy model, we prove functional limit theorems for the process \[Z_N(t)=\sum_{i=1}^N\mathrm{e}^{\xi_i(s_N+t)}\] as $N\to\infty$, where $s_N$ is a non-negative sequence converging to $+\infty$. The limiting process depends heavily on the growth rate of the sequence $s_N$. If $s_N$ grows slowly in the sense that $\liminf_{N\to\infty}\log N/s_N>\lambda_2$ for some critical value $\lambda_2>0$, then the limit is an Ornstein--Uhlenbeck process. However, if $\lambda:=\lim_{N\to\infty}\log N/s_N\in(0,\lambda_2)$, then the limit is a certain completely asymmetric $\alpha$-stable process $\mathbb {Y}_{\alpha ;\xi}$.