The weak convergence of regenerative processes using some excursion path decompositions

Lambert, Amaury; Simatos, Florian

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 2, pp. 492-511.

Nous considérons des processus régénératifs à valeurs dans un espace polonais quelconque. Nous définissons leurs excursions $\epsilon$-grandes comme les excursions $e$ telles que $\varphi \left(e\right)>\epsilon$, où $\varphi$ est une fonctionnelle donnée sur l’espace des excursions, qui peut par exemple être la longueur ou la hauteur de $e$. Nous établissons une condition générale garantissant la convergence d’une suite de processus régénératifs, qui porte sur la convergence des excursions $\epsilon$-grandes et de leurs extrémités, pour tout $\epsilon$ dans un ensemble dont l’adhérence contient $0$. Enfin, nous donnons plusieurs conditions suffisantes sur les mesures d’excursion de cette suite pour que cette condition générale soit satisfaite, et nous discutons de possibles généralisations de notre approche à certains processus pouvant être écrits comme la concaténation de motifs i.i.d.

We consider regenerative processes with values in some general Polish space. We define their $\epsilon$-big excursions as excursions $e$ such that $\varphi \left(e\right)>\epsilon$, where $\varphi$ 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 $\epsilon$-big excursions and of their endpoints, for all $\epsilon$ 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.

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