The weak convergence of regenerative processes using some excursion path decompositions

Amaury Lambert (1) , Florian Simatos (2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 TU/e - Eindhoven University of Technology [Eindhoven]

We consider regenerative processes with values in some general Polish space. We define their ε-big excursions as excursions e such that φ(e) > ε, where φ 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 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. motifs.

Mots clés: Excursion measure; Weak convergence; Queueing theory; Regenerative process; Excursion theory

Domaines: Probabilités [math.PR]

Amaury Lambert, Florian Simatos. The weak convergence of regenerative processes using some excursion path decompositions. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2014, 50 (2), pp.492-511. ⟨10.1214/12-AIHP531⟩. ⟨hal-01890571⟩