Bootstrap Central Limit Theorem for Chains of Infinite Order via Markov   Approximations
  We present a new approach to the bootstrap for chains of infinite order taking values on a finite alphabet. It is based on a sequential Bootstrap Central Limit Theorem for the sequence of canonical Markov approximations of the chain of infinite order. Combined with previous results on the rate of approximation this leads to a Central Limit Theorem for the bootstrapped estimator of the sample mean which is the main result of this paper. 
P. Collet, D. Duarte and A. Galves