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August 2011 Functional limit theorems for sums of independent geometric Lévy processes
Zakhar Kabluchko
Bernoulli 17(3): 942-968 (August 2011). DOI: 10.3150/10-BEJ299


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


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Zakhar Kabluchko. "Functional limit theorems for sums of independent geometric Lévy processes." Bernoulli 17 (3) 942 - 968, August 2011.


Published: August 2011
First available in Project Euclid: 7 July 2011

zbMATH: 1304.00020
MathSciNet: MR2817612
Digital Object Identifier: 10.3150/10-BEJ299

Keywords: $α$-stable processes , Functional limit theorem , Geometric Brownian motion , random energy model

Rights: Copyright © 2011 Bernoulli Society for Mathematical Statistics and Probability

Vol.17 • No. 3 • August 2011
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