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Institute of Mathematical Statistics, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2(50), 2014

DOI: 10.1214/12-aihp531

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The weak convergence of regenerative processes using some excursion path decompositions

Journal article published in 2012 by Amaury Lambert ORCID, Florian Simatos
This paper is made freely available by the publisher.
This paper is made freely available by the publisher.

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Abstract

We consider regenerative processes with values in some Polish space. We define their ε-big excursions as excursions e such that f(e)>ε, where f is some given functional on the space of excursions which can be thought of as, e.g., the length or the height of e. We establish a general condition that guarantees the convergence of a sequence of regenerative processes involving the convergence of ε-big excursions and of their endpoints, for all ε in a countable set whose closure contains 0. Finally, we provide various sufficient conditions on the excursion measures of this sequence for this general condition to hold and discuss possible generalizations of our approach to processes that can be written as the concatenation of i.i.d. paths. ; Comment: Final version accepted for publication in Annales de l'Institut Henri Poincaré