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Problèmes de recouvrement et points exceptionnels pour la marche aléatoire et le mouvement brownien

Zhan Shi (2004/2005)

Séminaire Bourbaki

La marche aléatoire (ou marche au hasard) est un objet fondamental de la théorie des probabilités. Un des problèmes les plus intéressants pour la marche aléatoire (ainsi que pour le mouvement brownien, son analogue dans un contexte continu) est de savoir comment elle recouvre des ensembles où se trouvent les points qui sont souvent (ou au contraire, rarement) visités, et combien il y a de tels points. Les travaux de Dembo, Peres, Rosen et Zeitouni permettent de résoudre plusieurs conjectures importantes...

Process level moderate deviations for stabilizing functionals

Peter Eichelsbacher, Tomasz Schreiber (2010)

ESAIM: Probability and Statistics


Functionals of spatial point process often satisfy a weak spatial dependence condition known as stabilization. In this paper we prove process level moderate deviation principles (MDP) for such functionals, which is a level-3 result for empirical point fields as well as a level-2 result for empirical point measures. The level-3 rate function coincides with the so-called specific information. We show that the general result can be applied to prove MDPs for various particular functionals, including...

Process-level large deviations for nonlinear Hawkes point processes

Lingjiong Zhu (2014)

Annales de l'I.H.P. Probabilités et statistiques

In this paper, we prove a process-level, also known as level-3 large deviation principle for a very general class of simple point processes, i.e. nonlinear Hawkes process, with a rate function given by the process-level entropy, which has an explicit formula.

Process-level quenched large deviations for random walk in random environment

Firas Rassoul-Agha, Timo Seppäläinen (2011)

Annales de l'I.H.P. Probabilités et statistiques

We consider a bounded step size random walk in an ergodic random environment with some ellipticity, on an integer lattice of arbitrary dimension. We prove a level 3 large deviation principle, under almost every environment, with rate function related to a relative entropy.

Product of exponentials and spectral radius of random k-circulants

Arup Bose, Rajat Subhra Hazra, Koushik Saha (2012)

Annales de l'I.H.P. Probabilités et statistiques

We consider n × n random k-circulant matrices with n → ∞ and k = k(n) whose input sequence {al}l≥0 is independent and identically distributed (i.i.d.) random variables with finite (2 + δ) moment. We study the asymptotic distribution of the spectral radius, when n = kg + 1. For this, we first derive the tail behaviour of the g fold product of i.i.d. exponential random variables. Then using this tail behaviour result and appropriate normal approximation techniques, we show that with appropriate scaling...

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