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Tight inequalities among set hitting times in Markov chains. (English) Zbl 1300.60083
Summary: Given an irreducible discrete time Markov chain on a finite state space, we consider the largest expected hitting time $$T(\alpha )$$ of a set of stationary measure at least $$\alpha$$ for $$\alpha \in (0,1)$$. We obtain tight inequalities among the values of $$T(\alpha )$$ for different choices of $$\alpha$$. One consequence is that $$T(\alpha ) \leq T(1/2)/\alpha$$ for all $$\alpha < 1/2$$. As a corollary, we have that, if the chain is lazy in a certain sense as well as reversible, then $$T(1/2)$$ is equivalent to the chain’s mixing time, answering a question of Y. Peres [Personal communication, 2012]. We furthermore demonstrate that the inequalities we establish give an almost everywhere pointwise limiting characterisation of possible hitting time functions $$T(\alpha )$$ over the domain $$\alpha \in (0,1/2]$$.

##### MSC:
 60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
##### Keywords:
Markov chains; hitting times
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##### References:
 [1] David J. Aldous, Some inequalities for reversible Markov chains, J. London Math. Soc. (2) 25 (1982), no. 3, 564 – 576. · Zbl 0489.60077 [2] David Aldous, László Lovász, and Peter Winkler, Mixing times for uniformly ergodic Markov chains, Stochastic Process. Appl. 71 (1997), no. 2, 165 – 185. · Zbl 0941.60080 [3] Graham Brightwell and Peter Winkler, Maximum hitting time for random walks on graphs, Random Structures Algorithms 1 (1990), no. 3, 263 – 276. · Zbl 0744.05044 [4] Navin Goyal, Problems from the AIM Workshop on Algorithmic Convex Geometry, accessed 11 February 2012: http://www.aimath.org/WWN/convexgeometry/convexgeometry.pdf, 2007. [5] David A. Levin, Yuval Peres, and Elizabeth L. Wilmer, Markov chains and mixing times, American Mathematical Society, Providence, RI, 2009. With a chapter by James G. Propp and David B. Wilson. · Zbl 1160.60001 [6] L. Lovász, Random walks on graphs: a survey, Combinatorics, Paul Erdős is eighty, Vol. 2 (Keszthely, 1993) Bolyai Soc. Math. Stud., vol. 2, János Bolyai Math. Soc., Budapest, 1996, pp. 353 – 397. · Zbl 0854.60071 [7] Roberto Imbuzeiro Oliveira, Mixing and hitting times for finite Markov chains, Electron. J. Probab. 17 (2012), no. 70, 12. · Zbl 1251.60059 [8] Yuval Peres, personal communication, 2012. [9] Yuval Peres and Perla Sousi, Mixing times are hitting times of large sets, electronically published in 2013, DOI 10.1007/s10959-013-0497-9. · Zbl 1323.60094 [10] -, personal communication, 2012.
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