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Tail Estimates for Sums of Variables Sampled by a Random Walk

Published online by Cambridge University Press:  01 March 2008

ROY WAGNER
ROY WAGNER*
Affiliation:
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-mail: rwagner@mta.ac.il)
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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.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 2 , March 2008 , pp. 307 - 316
Copyright
Copyright © Cambridge University Press 2007

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References

[1]Artstein-Avidan, S. and Milman, V. (2006) Logarithmic reduction of the level of randomness in some probabilistic geometric constructions. J. Funct. Anal. 235 297–329.CrossRefGoogle Scholar
[2]Bennett, G. (1962) Probability inequalities for the sum of independent random variables. J. Amer. Statist. Assoc. 57 33–45.CrossRefGoogle Scholar
[3]Dinwoodie, I. H. (1995) A probability inequality for the occupation measure of a reversible Markov chain. Ann. Appl. Probab. 5 37–43.CrossRefGoogle Scholar
[4]Gillman, D. (1998) A Chernoff bound for random walks on expander graphs. SIAM J. Comput. 27 1203–1220.CrossRefGoogle Scholar
[5]Hoory, S., Linial, N. and Wigderson, A. (2006) Expander graphs and their applications. Bulletin of the AMS 43 439–561.CrossRefGoogle Scholar
[6]Kargin, V. (2007) A large deviation inequality for vector functions on finite reversible Markov Chains (2007). Ann. Appl. Probab. 17 (4)1202–1221.CrossRefGoogle Scholar
[7]Leon, C. A. and Perron, F. (2004) Optimal Hoeffding bounds for discrete reversible Markov chains. Ann. Appl. Probab. 14 958–970.CrossRefGoogle Scholar
[8]Lezaud, P. (1998) Chernoff type bound for finite Markov chains. Ann. Appl. Probab. 8 849–867.CrossRefGoogle Scholar