Probabilities and Statistic seminar

The aim of this presentation is to show different results obtained in the field of semi-Markov processes focusing on risk/reliability indicators and related statistical inference aspects. The first part concerns discrete-time semi-Markov processes which are defined in a finite state space. Theoretical results are obtained in terms of evaluation and statistical estimation of risk indicators such as the mean time to failure. The asymptotic behavior of the empirical estimators is studied in the case of one single (large) trajectory. The results are illustrated on both real and simulated wind data. The second part of the presentation concerns partially observed semi-Markov chains such as the hidden Markov renewal and the hidden semi-Markov chains. Statistical estimation results are presented for reliability indicators such as the failure occurrence rate. In the third part, we present bootstrap estimators of risk indicators when the state space is discrete along with posterior concentration rates of the kernel density when the state space is continuous.

(Exposé en français avec des diapositives en anglais.)

Stochastic partial differential equations (SPDEs) are a major subject of current research in probability and analysis, with rich methodologies for studying existence and regularity. At the same time, SPDEs are increasingly used as statistical models for spatially and temporally structured data, where inference requires learning unknown parameters or functions from observations. In this talk, we consider Bayesian inference for the reaction function in a stochastic reaction-diffusion equation, based on a single solution trajectory observed continuously in space over a fixed time interval. We place a Gaussian process prior on the reaction function and derive posterior contraction rates in a novel asymptotic regime in which the spatial domain grows while the observation horizon remains fixed. In this setting, the SPDE solution becomes spatially ergodic and converges to a stationary process, which allows us to prove concentration inequalities for spatial averages of the solution. The proofs combine tools from Malliavin calculus – most notably the Clark–Ocone formula – with sharp bounds on the marginal densities of the SPDE.

Programme à venir.

Plus d’informations ici.

Exposé à Metz. Titre et résumé à venir.

Exposé à Metz. Titre et résumé à venir.