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The power of averaging at two consecutive time steps: proof of a mixing conjecture by Aldous and Fill. (English. French summary) Zbl 1382.60095

Summary: Let \((X_{t})_{t=0}^{\infty}\) be an irreducible reversible discrete-time Markov chain on a finite state space \(\Omega\). Denote its transition matrix by \(P\). To avoid periodicity issues (and thus ensuring convergence to equilibrium) one often considers the continuous-time version of the chain \((X_{t}^{\mathrm{c}})_{t\geq 0}\) whose kernel is given by \(H_{t}:=e^{-t}\sum_{k}(tP)^{k}/k!\). Another possibility is to consider the associated averaged chain \((X_{t}^{\mathrm{ave}})_{t=0}^{\infty}\), whose distribution at time \(t\) is obtained by replacing \(P^{t}\) by \(A_{t}:=(P^{t}+P^{t+1})/2\).{
} A sequence of Markov chains is said to exhibit (total-variation) cutoff if the convergence to stationarity in total-variation distance is abrupt. Let \((X_{t}^{(n)})_{t=0}^{\infty}\) be a sequence of irreducible reversible discrete-time Markov chains. In this work we prove that the sequence of associated continuous-time chains exhibits total-variation cutoff around time \(t_{n}\) iff the sequence of the associated averaged chains exhibits total-variation cutoff around time \(t_{n}\). Moreover, we show that the width of the cutoff window for the sequence of associated averaged chains is at most that of the sequence of associated continuous-time chains. In fact, we establish more precise quantitative relations between the mixing-times of the continuous-time and the averaged versions of a reversible Markov chain, which provide an affirmative answer to a problem raised by D. Aldous and J. Fill [Reversible Markov Chains and Random Walks on Graphs. Berkeley: University of California (2002)].

MSC:

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

[1] D. Aldous and J. Fill.Reversible Markov Chains and Random Walks on Graphs. University of California, Berkeley, 2002. Available athttps://www.stat.berkeley.edu/ aldous/RWG/book.pdf.
[2] N. Alon and J. H. Spencer.The Probabilistic Method. John Wiley & Sons, New York, 2004. · Zbl 1333.05001
[3] R. Basu, J. Hermon and Y. Peres. Characterization of cutoff for reversible Markov chains.Ann. Probab.45(3) (2017) 1448-1487. · Zbl 1374.60129
[4] G.-Y. Chen and L. Saloff-Coste. Comparison of cutoffs between lazy walks and Markovian semigroups.J. Appl. Probab.50(4) (2013) 943-959. · Zbl 1288.60088
[5] S. Griffiths, R. Kang, R. Oliveira and V. Patel. Tight inequalities among set hitting times in Markov chains.Proc. Amer. Math. Soc.142(2014) 3285-3298. · Zbl 1300.60083
[6] C. Le Merdy and Q. Xu. Maximal theorems and square functions for analytic operators on \(L^{p}\)-spaces.J. Lond. Math. Soc.86(2) (2012) 343-365. · Zbl 1264.47036
[7] D. A. Levin, Y. Peres and E. L. Wilmer.Markov Chains and Mixing Times. Amer. Mathematical Society, Providence, 2009. · Zbl 1160.60001
[8] E. Lubetzky and A. Sly. Cutoff phenomena for random walks on random regular graphs.Duke Math. J.153(3) (2010) 475-510. · Zbl 1202.60012
[9] Y. Peres and P. Sousi. Mixing times are hitting times of large sets.J. Theoret. Probab.28(2) (2015) 488-519. · Zbl 1323.60094
[10] E. M. Stein. On the maximal ergodic theorem.Proc. Natl. Acad. Sci. USA47(12) (1961) 1894. · Zbl 0182.47102
[11] D. V. Widder.An Introduction to Transform Theory, Vol. 42. Academic Press, San Diego, 1971. · Zbl 0219.44001
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