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

TL;DR

This paper develops second-order Edgeworth expansions for transition densities of nonhomogeneous Markov chains converging to diffusion processes, especially for small time lags, with applications in statistical density approximations.

Contribution

It introduces novel second-order Edgeworth expansions for nonhomogeneous Markov chains with small time lags, extending previous results to more general diffusion limits.

Findings

Established second-order Edgeworth expansions for transition densities.

Extended analysis to nonhomogeneous diffusion limits.

Provided small time asymptotics relevant for statistical applications.

Abstract

We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Second order Edgeworth type expansions for transition densities are proved. The paper differs from recent results in two respects. We allow nonhomogeneous diffusion limits and we treat transition densities with time lag converging to zero. Small time asymptotics are motivated by statistical applications and by resulting approximations for the joint density of diffusion values at an increasing grid of points.