Laboratoire de Mathématiques Pures et Appliquées
Joseph Liouville

Groupe de travail de Probabilités, statistiques, théorie ergodique du 1er octobre

Alexandre Chotard (ULCO, LISIC)

I will present new conditions which allow to easily prove the irreducibility and aperiodicity of a Markov chain, which are useful properties to show the existence of an invariant measure, the ergodicity of the chain, and apply theorems such as a law of large numbers.

For a Markov chain $(\Phi_k)$ modelled by a function $F$ and an i.i.d. sequence $(U_k)$ such that $\Phi_{k+1}=F(\Phi_k, \alpha(\Phi_k,U_{k+1}))$,
under some conditions on $F$ and $\alpha$, irreducibility and aperiodicity can be proven by analyzing deterministic sequences. Our work extends previous similar conditions for irreducibility which impose the full function $(x,u)\mapsto F(x, \alpha(x,u))$ to be $C^{\infty}$. In contrast, we assume $F$ to be $C^1$, and the function $(x,u)\mapsto F(x, \alpha(x,u))$ can be discontinuous. The condition for aperiodicity is new. In both cases, the conditions we propose are proven to be necessary and sufficient.

We provide in this presentation the necessary background to express these conditions, and present non-trivial examples in which we prove irreducibility and aperiodicity. Joint work with Anne Auger.

Informations : 15:00 - 16:00 Salle C115