Les chaînes de Markov à mémoire de longueur variable sont des sources probabilistes pour lesquelles la production d'une lettre dépend d'un passé fini, mais dont la longueur dépend du temps est n'est pas bornée. Elles sont définies à partir d'un arbre T qui est un sous-arbre de l'arbre de tous les mots. Contrairement aux chaînes de Markov d'ordre fini standard, ces sources n'admettent pas toujours de mesure de probabilité stationnaire, ou peuvent en admettre plusieurs. La forme de l'arbre T joue un rôle essentiel dans cette affaire. On montrera quelques outils adaptés à la question et, sous certaines hypothèses, on donnera une CNS d'existence et d'uniciteé d'une telle mesure de probabilité.
Travail en collaboration avec P. Cénac, B. Chauvin et F. Paccaut.[-]

Traditionally homogenization asks whether average behavior can be discerned from Hamilton-Jacobi equations that are subject to high-frequency fluctuations in spatial variables. A similar question can be asked for the associated Hamiltonian ODEs. When the Hamiltonian function is convex in momentum variable, these two questions turn out to be equivalent. This equivalence breaks down for general Hamiltonian functions. In this talk I will give a dynamical system formulation for homogenization and address some result concerning weak and strong homogenization phenomena.[-]

In this paper we study asymptotic properties of random forests within the framework of nonlinear time series modeling. While random forests have been successfully applied in various fields, the theoretical justification has not been considered for their use in a time series setting. Under mild conditions, we prove a uniform concentration inequality for regression trees built on nonlinear autoregressive processes and, subsequently, use this result to prove consistency for a large class of random forests. The results are supported by various simulations. (This is joint work with Mikkel Slot Nielsen.)[-]

The class of integer-valued trawl processes has recently been introduced for modelling univariate and multivariate integer-valued time series with short or long memory.

In this talk, I will discuss recent developments with regards to model estimation, model selection and forecasting of such processes. The new methods will be illustrated in an empirical study of high-frequency financial data.

This is joint work with Mikkel Bennedsen (Aarhus University), Asger Lunde (Aarhus University) and Neil Shephard (Harvard University).[-]

In this talk, I will present mixing properties for a broad class of Poisson count time series satisfying a certain contraction condition. Using specific coupling techniques, we obtain absolute regularity at a geometric rate not only for stationary Poisson-GARCH processes but also for models with an explosive trend. Easily verifiable sufficient conditions for absolute regularity can be deduced from our general results for a variety of models including classical (log-)linear models.[-]