Shifting processes with cyclically exchangeable increments at random

Loïc Chaumont (1) , Gerónimo Uribe Bravo (2)

1 LAREMA - Laboratoire Angevin de Recherche en Mathématiques
2 Instituto de Investigaciones en Matematicas Aplicadas y en Sistemas

We propose a path transformation which applied to a cyclically exchangeable increment process conditions its minimum to belong to a given interval. This path transformation is then applied to processes with start and end at 0. It is seen that, under simple conditions, the weak limit as $\varepsilon\rightarrow0$ of the process conditioned on remaining above $-\varepsilon$ exists and has the law of the Vervaat transformation of the process. We examine the consequences of this path transformation on processes with exchangeable increments, Lévy bridges, and the Brownian bridge.

Mots clés: Cyclic exchangeability; Vervaat transformation; Brownian bridge; three dimensional Bessel bridge; uniform law; path transformation; occupation time.; occupation time

Domaines: Probabilités [math.PR]

Loïc Chaumont, Gerónimo Uribe Bravo. Shifting processes with cyclically exchangeable increments at random. 2014. ⟨hal-01023079⟩