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We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Local limit theorems for transition densities are proved.

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In this paper, we consider projection estimates for Lévy densities in high-frequency setup. We give a unified treatment for different sets of basis functions and focus on the asymptotic properties of the maximal deviation distribution for these estimates. Our results are based on the idea to reformulate the problems in terms of Gaussian processes of some special type and to further analyze these Gaussian processes. In particular, we construct a sequence of excursion sets, which guarantees the convergence of the deviation distribution to the Gumbel distribution. We show that the rates of convergence presented in previous articles on this topic are logarithmic and construct the sequences of accompanying laws, which approximate the deviation distribution with polynomial rate.MSC 2010 subject classifications: Primary 60G51, 62M99; secondary 62G05.

We study the sensitivity of the densities of non degenerate diffusion processes and related Markov Chains with respect to a perturbation of the coefficients. Natural applications of these results appear in models with misspecified coefficients or for the investigation of the weak error of the Euler scheme with irregular coefficients. Résumé. Nousétudions la sensibilité des densités de processus de diffusion non dégénérés et desChaînes de Markov associées par rapportà une perturbation des coefficients. Ces résultats trouvent des applications naturelles dans l'étude de modèles avec incertitude sur les coefficients ou pour l'analyse de l'erreur faible du schéma d'Eulerà coefficients irréguliers.