Tail Estimates for Sums of Variables Sampled by a Random Walk

We prove tail estimates for variables of the form ∑ i f(Xi), where (Xi)i is the trajectory of a random walk on an undirected graph (or, equivalently, a reversible Markov chain). The estimates are in terms of the range of the function f , its variance, and the spectrum of the graph. The purpose of our estimates is to determine the number of random walk generated samples which are required for approximating the expectation of a distribution on the vertices of a graph, especially an expander. The estimates must therefore provide information for fixed number of samples (as in Gillman’s [4]) rather than just asymptotic information. Our proofs are more elementary than other proofs in the literature, and our results are sharper. We obtain Bernstein and Bennett-type inequalities, as well as an inequality for subgaussian variables.