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

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.

Weak error for Continuous Time Markov Chains related to fractional in time P(I)DEs

“…The exponential control is derived similarly to the diffusive case, and comes from the specific form of the parametrix expansion used to analyze the error. The control in small time can also be derived from this representation, similarly to what occurs in [KM09]. To make this last point clear we give some details in Appendix A.…”

“…for which the innovations are not necessarily Gaussian, to enter a Gaussian asymptotics specified by the Gaussian LLT, a certain number of time steps is needed. This is why we impose the condition t i ≥ h δ , δ < 1/5 which is the one required in [KM09] (see assumption (B2) therein) to establish the Gaussian LLT in short time. The behaviour of the constants in large time can also be derived from [KM00] (similarly to the procedure presented in Appendix A.1).…”

“…The behaviour of the constants in large time can also be derived from [KM00] (similarly to the procedure presented in Appendix A.1). For the control in short time we refer to Remark 1 after Theorem 1 in [KM09]. Anyhow, for the reader's convenience, we recall in Appendix A.1 the approach developed therein.…”

“…The remainder can be handled similarly (see[KM02]). This analysis can be performed as well for the Markov Chain approximation, see[KM09].A.2. Strictly Stable Case.…”

Statistical convergence of Markov experiments to diffusion limits

“…We expect that such results could be shown by using expansions for transition densities of Markov random walks. The approach of this paper is based on expansions developed in [12]. The latter paper only considers Markov chains with continuous state space.…”