Functional limit theorems for sums of independent geometric Lévy processes

Abstract

Let ${\xi }_i$, $i\in \mathbb{N}$, be independent copies of a Lévy process $\{\xi (t),t\ge 0\}$. Motivated by the results obtained previously in the context of the random energy model, we prove functional limit theorems for the process \[Z_N(t)=\sum_{i=1}^N\mathrm{e}^{ξ_i(s_N+t)}\] 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 ${\mathrm{liminf}}_{N\to \infty }\log N/s_N>{\lambda }_2$ 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 ${\mathbb{Y}}_{\alpha ;\xi }$.

Published in at http://dx.doi.org/10.3150/10-BEJ299 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (3)

• Functional limit theorems for sums of independent geometric Lévy processes
• Limiting Distributions for Sums of Independent Random Products
• Limit laws for sums of independent random products: the lattice case

Cited by in corpus (6)

• An exceptional max-stable process fully parameterized by its extremal coefficients
• Complex Random Energy Model: Zeros and Fluctuations
• Functional limit theorems for sums of independent geometric Lévy processes
• Generalized Random Energy Model at Complex Temperatures
• Limit laws for sums of independent random products: the lattice case
• Max-stable processes and stationary systems of Lévy particles