Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

The goal of this presentation is to discuss the reliable sequential detection of transient changes in a multi-parameter exponential distribution. The sequentially observed data are represented by a sequence of independent random vectors with the exponentially distributed components.

We first consider a multidimensional diffusion with jumps driven by a Brownian motion and a Poisson random measure associated with a Lévy process without Gaussian component, whose drift coefficient depends on a multidimensional unknown parameter. We prove the local asymptotic normality (LAN) property from high-frequency discrete observations with increasing observation window by assuming some hypotheses on the coefficients of the equation, the ergodicity of the solution and the integrability of the Lévy measure.

Firstly, I will present the definition of spectral disjointness of two automorphisms and show a criterion which can be used  to distinguish this property. Secondly, I will show how to see that powers of a generic interval exchange transformation of three intervals satisfy the asumptions of the criterion.

The result is based on a joint work with Adam Kanigowski.

Nous allons donner des conditions suffisantes pour la loi des logarithmes itérés bornée
pour des champs aléatoires, en considérant la sommation sur des rectangles. Nous nous concentrons sur
deux classes de champs aléatoires strictement stationnaires : les martingales pour l'ordre lexicographique
et les orthomartingales.

Threshold diffusions are regime-switching continuous-time models where the model switches between two or more different model behaviors as the state variable cross certain domains.  We consider the statistical estimation on the switching threshold value for a discretely observed one-dimensional diffusion process.

We introduce a parametrised family of maps $\{S_{\eta}\}_{\eta \in [1,2]}$, called symmetric doubling maps, defined on $[-1,1]$ by $S_\eta (x)=2x-d\eta$, where $d\in \{-1,0,1 \}$. Each map $S_\eta$ generates binary expansions with digits $-1$, 0 and 1. We study the frequency of the digit 0 in typical expansions as a function of the parameter $\eta$. The transformations $S_\eta$ have a natural ergodic invariant measure $\mu_\eta$ that is absolutely continuous with respect to Lebesgue measure.

In this talk, we  study  the probabilistic behavior of particular trajectories (finite or periodic) of a  given dynamical system.
For these particular trajectories, ergodic theorems  do  not apply, 
and we  explain the main principles  of  the Dynamical Analysis Method, in the case of the Gauss map.
In this case, finite trajectories  coincide with rational trajectories, and  thus executions of the Euclid algorithm.