Theorem 5.2 for All Distributions And

equivalent to saying that states can be represented by points in some euclidean space so that the distance between i and j is p pij.) It is virtually never possible to nd a blind rule for generating which is mean-optimal. If we consider only stopping at some xed time, even approximating may take far more time than a mean-optimal rule (for example when the chain is almost periodic). However, the following theorem shows that we can get within " of (from below) with a fast blind rule. Let " > 0, put k = dlog(1=")e, and let t = 8dTmixe. Let Z be the sum of k independent random variables Y1 blind rule which stops the chain at time Z. Theorem 7.2 For any starting distribution , the rule ?" produces a distribution satisfying (1?"); and has mean length O(Tmix log(1=")). We note that similar convergence speed can be achieved using other \averaging" rules, e.g., the Poisson stopping time of Aldous 2]. 8 Exact Mixing in an Unknown Chain In the previous section we saw that there is at least one practical stopping rule for approximating the stationary distribution when all that is known about the chain is its mixing time (or a good upper bound for it). Even this requirement can be circumvented: Aldous 4] gives a rule that comes within total variation " of the stationary distribution in expected time O(h=" 2 where h = maxi;j H(i;j) is the maximum hitting time of the chain, provided the number n of states is known. Amazingly, one can even achieve the the stationary distribution exactly in an unknown chain. A stopping rule for doing so is described in Asmussen et al. 5], but it is complex and requires perfect generation of random variables with certain exponential distributions. The expected number of steps required appears to be super-polynomial in the maximum hitting time, although no bound or estimate is given in the paper. It turns out, however, that there is a simple stopping rule ? which can reach the stationary distribution exactly, in any irreducible, n-state Markov chain; the rule nishes in time polynomial in h, requires only coin-ips for its ran-domization, and can even be made deterministic unless the chain itself is completely deterministic. We describe ? as follows. Given a state i and a positive integer t, a t-exit of i consists of waiting X steps after being in …