Functional limit theorems for sums of independent geometric Lévy processes

Abstract

Let $ξ_i, i ∈ ℕ$, be independent copies of a Lévy process {$ξ(t), t ≥ 0$}. 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 $N → ∞$, where $s_N$ is a non-negative sequence converging to $+∞$. The limiting process depends heavily on the growth rate of the sequence $s_N$. If $s_N$ grows slowly in the sense that $\lim \inf _{N→∞ } \log N / s_N > λ_2$ for some critical value $λ_2 > 0$, then the limit is an Ornstein–Uhlenbeck process. However, if $λ := \lim _{N→∞ }\log N /s_N ∈ (0, λ_2)$, then the limit is a certain completely asymmetric $α$-stable process $\mathbb{Y}_{\alpha ;\xi}$.