Stein’s method and non-reversible Markov chains

Abstract

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.