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Large deviations for the empirical measure of heavy-tailed Markov renewal processes

Part of: Limit theorems Special processes

Published online by Cambridge University Press:  19 September 2016

Mauro Mariani  and
Lorenzo Zambotti
Mauro Mariani*
Affiliation:
Università degli Studi di Roma La Sapienza
Lorenzo Zambotti*
Affiliation:
Université Paris 6 ‒ Pierre et Marie Curie
*
* Postal address: Dipartimento di Matematica, Università degli Studi di Roma La Sapienza, Piazzale Aldo Moro 5, 00185, Roma, Italy. Email address: mariani@mat.uniroma1.it
** Postal address: Laboratoire de Probabilités et Modèles Aléatoires (CNRS UMR. 7599), Université Paris 6 ‒ Pierre et Marie Curie, U.F.R. Mathématiques, Case 188, 4 place Jussieu, 75252 Paris cedex 05, France. Email address: lorenzo.zambotti@upmc.fr
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Abstract

A large deviations principle is established for the joint law of the empirical measure and the flow measure of a Markov renewal process on a finite graph. We do not assume any bound on the arrival times, allowing heavy-tailed distributions. In particular, the rate function is in general degenerate (it has a nontrivial set of zeros) and not strictly convex. These features show a behaviour highly different from what one may guess with a heuristic Donsker‒Varadhan analysis of the problem.

Keywords

Large deviationempirical measureMarkov renewal processheavy tail

MSC classification

Primary: 60F10: Large deviations 60K15: Markov renewal processes, semi-Markov processes
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 3 , September 2016 , pp. 648 - 671
Copyright
Copyright © Applied Probability Trust 2016 

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