Statistical convergence of Markov experiments to diffusion limits

Abstract

Assume that one observes the $k\text{th},2k\text{th},\ldots,nk\text{th}$ value of a Markov chain $X_{1,h},\ldots ,X_{nk,h}$. That means we assume that a high frequency Markov chain runs in the background on a very fine time grid but that it is only observed on a coarser grid. This asymptotics reflects a set up occurring in the high frequency statistical analysis for financial data where diffusion approximations are used only for coarser time scales. In this paper, we show that under appropriate conditions the $\mathrm{L}_{1}$-distance between the joint distribution of the Markov chain and the distribution of the discretized diffusion limit converges to zero. The result implies that the LeCam deficiency distance between the statistical Markov experiment and its diffusion limit converges to zero. This result can be applied to Euler approximations for the joint distribution of diffusions observed at points $\Delta,2\Delta,\ldots ,n\Delta$. The joint distribution can be approximated by generating Euler approximations at the points $\Delta k^{-1},2\Delta k^{-1},\ldots ,n\Delta$. Our result implies that under our regularity conditions the Euler approximation is consistent for $n\to\infty$ if $nk^{-2}\to0$.