## 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:

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