Functional limit theorems for sums of independent geometric Lévy processes

Details Functional limit theorems for sums of independent geometric Lévy processes Functional limit theorems for sums of independent geometric Lévy processes

2009-11-21

arxiv.org/abs/0911.4139v2

Bernoulli 2011, Vol. 17, No. 3, 942-968

Mathematics

Probability

Published in at http://dx.doi.org/10.3150/10-BEJ299 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/10-BEJ299

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

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