Functional limit theorems for sums of independent geometric Lévy processes

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 $\mathbb {Y}_{\alpha ;\xi}$.