Stretched random walks and the behaviour of their summands.

Michel Broniatowski (1) , Zhansheng Cao (1)

1 LSTA - Laboratoire de Statistique Théorique et Appliquée

This paper explores the joint behaviour of the summands of a random walk when their mean value goes to infinity as its length increases. It is proved that all the summands must share the same value, which extends previous results in the context of large exceedances of finite sums of i.i.d. random variables. Some consequences are drawn pertaining to the local behaviour of a random walk conditioned on a large deviation constraint on its end value. It is shown that the sample paths exhibit local oblic segments with increasing size and slope as the length of the random walk increases.

Mots clés: random walk; extreme deviation; Gibbs Principle

Domaines: Statistiques [math.ST]; Théorie [stat.TH]; Probabilités [math.PR]

Michel Broniatowski, Zhansheng Cao. Stretched random walks and the behaviour of their summands.. 2012. ⟨hal-00701807⟩