Some Continuous Edgeworth Expansions for Markov Chains with Applications to Bootstrap

This paper deals with the first order Edgeworth expansions for sums related to an ergodic Markov chain with general state space. In the first part of the paper, we establish certain continuity, w.r.t. the transition probability function and the initial distribution, in these expansions. In the second part, we illustrate the use of our continuous expansions in the area of bootstrap. We consider bootstrapping the distribution of the (sample) mean of a fixed real function of a Markov chain. Under a conditional non-latticeness condition, the bootstrap is shown to be second order accurate. As a second application we obtain Edgeworth expansions for the bootstrap approximation to the sampling distribution of the m.l.e. of a particular transition probability in a finite Markov chain. It is shown that the bootstrap is second order accurate and is therefore superior to the normal approximation, if the transition probability is irrational. In the other case, the exact asymptotic upper bound constant in the O(n−12) rate of bootstrap approximation is determined.