Some Specialized Comparison Techniques for Doubly Stochastic Finite Markov Chains

This paper considers certain comparison techniques involving Markov chains with transition matrices Pa = aI+(1?a)P where P is the transition matrix of a doubly stochastic Markov chain. This paper provides upper bounds for how far the Markov chain with transition matrix Pb is from uniformly distributed after n steps. These upper bounds involve how far the Markov chain with transition matrix Pa is from uniformly distributed after m steps and probabilities involving binomial random variables, m, n, a, and b. These upper bounds are provided for cases where b > a and for cases where b < a. The paper also provides an example involving random walks on nite groups where the result in the case b < a is useful.