Local limit theorems on the convergence of Markov chains to diffusion processes

One proves the uniform convergence of the densities of the finite-dimensional distributions of certain families of Markov chains to the densities of the finite-dimensional distributions of a nondegenerate diffusion process.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 277–292, 1986.The author is grateful to S. A. Molchanov for numerous useful discussions.