In this paper we derive non asymptotic deviation bounds for
$${\P}_{\nu}\left(\right|\frac{1}{t}{\int}_{0}^{t}V\left({X}_{s}\right)\mathrm{d}s-\int V\mathrm{d}\mu |\ge R)$$
where $X$ is a $\mu $ stationary and ergodic Markov process and $V$ is some $\mu $ integrable function. These bounds are obtained under various moments assumptions for $V$, and various regularity assumptions for $\mu $. Regularity means here that $\mu $ may satisfy various functional inequalities (F-Sobolev, generalized Poincaré etc.).

This paper presents different approaches, based on functional inequalities, to study the speed of convergence in total variation distance of ergodic diffusion processes with initial law satisfying a given integrability condition. To this end, we give a general upper bound “à la Pinsker” enabling us to study our problem firstly via usual functional inequalities (Poincaré inequality, weak Poincaré,…) and truncation procedure, and secondly through the introduction of new functional inequalities ${\mathcal{I}}_{\psi}$....

In this paper we derive non asymptotic deviation bounds for
$${\mathbb{P}}_{\nu}\left(\right|\frac{1}{t}{\int}_{0}^{t}V\left({X}_{s}\right)\mathrm{d}s-\int V\mathrm{d}\mu |\ge R)$$ where is a stationary and ergodic Markov process and is some integrable function. These bounds are obtained under various moments assumptions for , and various regularity assumptions for . Regularity means here that may satisfy various functional inequalities (F-Sobolev,
generalized Poincaré etc.).

This paper studies limit theorems for Markov chains with general state space under conditions which imply subgeometric ergodicity. We obtain a central limit theorem and moderate deviation principles for additive not necessarily bounded functional of the Markov chains under drift and minorization conditions which are weaker than the Foster–Lyapunov conditions. The regeneration-split chain method and a precise control of the modulated moment of the hitting time to small sets are employed in the proof....

We consider a Vlasov-Fokker-Planck equation governing the evolution
of the density of interacting and diffusive matter in the space of
positions and velocities.
We use a probabilistic interpretation to obtain convergence towards
equilibrium in Wasserstein distance with an explicit exponential
rate. We also prove a propagation of chaos property for an
associated particle system, and give rates on the approximation of
the solution by the particle system. Finally, a transportation
inequality...

We investigate here the central limit theorem of the increment ratio statistic of a multifractional Brownian motion, leading to a CLT for the time varying Hurst index. The proofs are quite simple relying on Breuer–Major theorems and an original strategy. A simulation study shows the goodness of fit of this estimator.

Equivalence of the spectral gap, exponential integrability of hitting times and Lyapunov conditions is well known. We give here the correspondence (with quantitative results) for reversible diffusion processes. As a consequence, we generalize results of Bobkov in the one dimensional case on the value of the Poincaré constant for log-concave measures to superlinear potentials. Finally, we study various functional inequalities under different hitting times integrability conditions (polynomial,…)....

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