Tail estimates for sums of variables sampled by a random walk

  • Authors:

  • Roy Wagner

  • Affiliations:

  • Computer science department, academic college of tel aviv–yaffa, 2 rabenu yerham st, jaffa 68182, israel and school of mathematical sciences, tel aviv university, tel aviv 69978, israel (e-m ...

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2008

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Abstract

We prove tail estimates for variables of the form ∑if(Xi), where (Xi)i is a sequence of states drawn from a reversible Markov chain, or, equivalently, from a random walk on an undirected graph. 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 chain/walk samples which are required for approximating the expectation of a distribution on 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 sub-Gaussian variables.