The method of exchangeable pairs has emerged as an important tool in proving limit theorems for Poisson, normal and other classical approximations. Here the method is used in a simulation context. We estimate transition probabilitites from the simulations and use these to reduce variances. Exchangeable pairs are used as control variates.

Finally, a general approximation theorem is developed that can be complemented by simulations to provide actual estimates of approximation errors.

Charles Stein has introduced a general approach to proving approximation theorems in probability theory. The method is being actively used for normal and Poisson approximation. This paper uses the method to derive rates of convergence of some simple Markov chains to their stationary distribution. The main purpose is to present Stein’s general approach in a simple setting where the many choices can be examined and compared.

This article presents a review of Stein’s method applied to the case of discrete random variables. We attempt to complete one of Stein’s open problems, that of providing a discrete version for chapter 6 of his book. This is illustrated by first studying the mechanics of comparison between two distributions whose characterizing operators are known, for example the binomial and the Poisson. Then the case where one of the distributions has an unknown characterizing operator is tackled. This is done for the hypergeometric which is then compared to a binomial. Finally the general case of the comparison of two probability distributions that can be seen as the stationary distributions of two birth and death chains is treated and conditions of the validity of the method are conjectured.

Let $W(\pi)$ be either the number of descents or inversions of a permutation $\pi \in S_n$. Stein's method is applied to show that $W$ satisfies a central limit theorem with error rate $n^{-1/2}$. The construction of an exchangeable pair $(W,W')$ used in Stein's method is non-trivial and uses a non-reversible Markov chain.

The antivoter model is a Markov chain on regular graphs which has a unique stationary distribution, but is not reversible. This makes the stationary distribution difficult to describe. Despite the fact that in general we know nothing about the stationary distribution other than it exists and is unique, we present a method for sampling exactly from this distribution. The method has running time $O(n^3 r / c)$, where $n$ is the number of nodes in the graph, $c$ is the size of the minimum cut in the graph, and $r$ is the degree of each node in the graph. We also show that the original chain has $O(n^3 r /c)$ mixing time. For the antivoter model on the complete graph we derive a closed form solution for the stationary distribution. Moreover we bound the total variation distance between the stationary distribution for the antivoter model on a multipartite graph and the stationary distribution on the complete graph, using Stein's method. Finally, we present computational experiments comparing the empirical Stein's method for estimating the stationary distribution to the classical frequency estimate.

This paper gives new proofs for many known results about the convergence in law of the bootstrap distribution to the true distribution of smooth statistics, whether the samples studied come from independent realizations of a random variable or dependent realizations with weak dependence. Moreover it suggests a novel bootstrap procedure and provides a proof that this new bootstrap works under uniform local dependence. The techniques employed are based on Stein’s method for empirical processes as developed by Reinert (1994). A weak law of large numbers for empirical measures via Stein’s method and applications. The last section provides some simulations and applications for which the relevant matlab functions are available from the first author.