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Limit theorems for some continuous-time random walks

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  01 July 2016

M. Jara  and
T. Komorowski
M. Jara*
Affiliation:
Université de Paris Dauphine and Instituto de Matématica Pura e Aplicada
T. Komorowski*
Affiliation:
Uniwersytet Marii Curie-Skłodowskiej and Polish Academy of Sciences
*
∗ Postal address: CEREMADE, UMR CNRS 7534, Université de Paris Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France. Email address: jara@ceremade.dauphine.fr
∗∗ Postal address: Institute of Mathematics, Uniwersytet Marii Curie-Skłodowskiej, pl. Marii Curie-Skłodowskiej 1, Lublin 20-031, Poland. Email address: komorow@hektor.umcs.lublin.pl
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Abstract

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In this paper we consider the scaled limit of a continuous-time random walk (CTRW) based on a Markov chain {Xn, n ≥ 0} and two observables, τ(∙) and V(∙), corresponding to the renewal times and jump sizes. Assuming that these observables belong to the domains of attraction of some stable laws, we give sufficient conditions on the chain that guarantee the existence of the scaled limits for CTRWs. An application of the results to a process that arises in quantum transport theory is provided. The results obtained in this paper generalize earlier results contained in Becker-Kern, Meerschaert and Scheffler (2004) and Meerschaert and Scheffler (2008), and the recent results of Henry and Straka (2011) and Jurlewicz, Kern, Meerschaert and Scheffler (2010), where {Xn, n ≥ 0} is a sequence of independent and identically distributed random variables.

Keywords

Continuous-time random walkMarkov chainsubordinatorjump process

MSC classification

Primary: 60G50: Sums of independent random variables; random walks 60F17: Functional limit theorems; invariance principles
Secondary: 60G18: Self-similar processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 3 , September 2011 , pp. 782 - 813
Copyright
Copyright © Applied Probability Trust 2011 

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