Functional limit theorems for sums of independent geometric Lévy processes

Abstract

Let $ξ_i$, $i\in \mathbb {N}$, be independent copies of a Lévy process $\{ξ(t),t\geq0\}$. Motivated by the results obtained previously in the context of the random energy model, we prove functional limit theorems for the process \[Z_N(t)=\sum_{i=1}^N\mathrm{e}^{ξ_i(s_N+t)}\] 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 $\liminf_{N\to\infty}\log N/s_N>λ_2$ for some critical value $λ_2>0$, then the limit is an Ornstein--Uhlenbeck process. However, if $λ:=\lim_{N\to\infty}\log N/s_N\in(0,λ_2)$, then the limit is a certain completely asymmetric $α$-stable process $\mathbb {Y}_{α;ξ}$.