Local limit theorems for transition densities of Markov chains converging to diffusions

Abstract

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

Small time Edgeworth-type expansions for weakly convergent nonhomogeneous Markov chains

“…Konakov and Mammen [19] obtained a nonuniform rate of convergence for the difference p h (0, T, x, ·) − p (0, T, x, ·) as n → ∞ in the case T 1. Edgeworth type expansions for the case T 1 and homogenous diffusions were obtained in [21].…”

“…For a short exposition of the parametrix method, see Sect. 3 and [19]. Similar representations hold for discrete time Markov chains X k,h , see Lemma 4 below.…”

“…Similar representations hold for discrete time Markov chains X k,h , see Lemma 4 below. The parametrix method for Markov chains was developed in [19] and it is exposed in Sect. 3.2.…”

“…Local limit theorems for Markov chains were given in [18][19][20]. In [19] it was shown that the transition density of a Markov chain converges with rate O(n −1/2 ) to the transition density in the diffusion model. For the proof there an analytical approach was chosen that made essential use of the parametrix method.…”

“…In Sect. 3.2 we will recall the parametrix approach developed in [19] for Markov chains. Technical discussions, auxiliary results and proofs are given in Sects.…”