# Functional limit theorems for sums of independent geometric Lévy processes - Mathematics > Probability

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Abstract: Let $\xi i$, $i\in \mathbb {N}$, be independent copies of a L\-{e}vy process$\{\xit,t\geq0\}$. Motivated by the results obtained previously in thecontext of the random energy model, we prove functional limit theorems for theprocess \Z Nt=\sum {i=1}^N\mathrm{e}^{\xi is N+t}\ as $N\to\infty$, where$s N$ is a non-negative sequence converging to $+\infty$. The limiting processdepends heavily on the growth rate of the sequence $s N$. If $s N$ grows slowlyin the sense that $\liminf {N\to\infty}\log N-s N>\lambda 2$ for some criticalvalue $\lambda 2>0$, then the limit is an Ornstein-Uhlenbeck process. However,if $\lambda:=\lim {N\to\infty}\log N-s N\in0,\lambda 2$, then the limit is acertain completely asymmetric $\alpha$-stable process $\mathbb {Y} {\alpha;\xi}$.

Author: ** Zakhar Kabluchko**

Source: https://arxiv.org/