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Surprise Probabilities in Markov Chains

Part of: Markov processes

Published online by Cambridge University Press:  16 March 2017

JAMES NORRIS ,
YUVAL PERES  and
ALEX ZHAI
JAMES NORRIS
Affiliation:
Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail: james@statslab.cam.ac.uk)
YUVAL PERES
Affiliation:
Microsoft Research, Redmond, Washington, WA 98052, USA (e-mail: peres@microsoft.com)
ALEX ZHAI
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, USA (e-mail: azhai@stanford.edu)
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Abstract

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In a Markov chain started at a state x, the hitting time τ(y) is the first time that the chain reaches another state y. We study the probability $\mathbb{P}_x(\tau(y) = t)$ that the first visit to y occurs precisely at a given time t. Informally speaking, the event that a new state is visited at a large time t may be considered a ‘surprise’. We prove the following three bounds.

  • • In any Markov chain with n states, $\mathbb{P}_x(\tau(y) = t) \le {n}/{t}$ .

  • • In a reversible chain with n states, $\mathbb{P}_x(\tau(y) = t) \le {\sqrt{2n}}/{t}$ for $t \ge 4n + 4$ .

  • • For random walk on a simple graph with n ≥ 2 vertices, $\mathbb{P}_x(\tau(y) = t) \le 4e \log(n)/t$ .

We construct examples showing that these bounds are close to optimal. The main feature of our bounds is that they require very little knowledge of the structure of the Markov chain.

To prove the bound for random walk on graphs, we establish the following estimate conjectured by Aldous, Ding and Oveis-Gharan (private communication): for random walk on an n-vertex graph, for every initial vertex x,

$$\sum_y \biggl( \sup_{t \ge 0} p^t(x, y) \biggr) = O(\log n). $$

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 4 , July 2017 , pp. 603 - 627
Copyright
Copyright © Cambridge University Press 2017 

References

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