Exponential approximation and Stein’s method of exchangeable pairs

The purpose of this paper is to further develop exponential approximation, using Stein’s method of exchangeable pairs. The first use of exchangeable pairs in exponential approximation was in the paper Chatterjee et al. (2011), which studied the spectrum of the Bernoulli-Laplace Markov chain. Unfortunately the results in Chatterjee et al. (2011), which use the Kolmogorov metric, are very complicated and seem quite hard to apply in other examples. We provide approximation results similar to those in Chatterjee et al. (2011), but which are significantly easier to compute. In particular, we do not see how to apply the results of Chatterjee et al. (2011) to the example in this paper. We work in a “smooth” test function metric, but also provide bounds in the Kolmogorov metric, which is defined for random variables W and Z to be