Convergence Times for Parallel Markov Chains

Lachaud, Béatrice; Ycart, Bernard

doi:10.1007/3-540-34774-7_22

Cited by 3 publications

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“…(t) = ℓ ,denote the first return time to ℓ after s E T These times are well defined: since, as we assume the chain to be positively recurent, every mean hitting time is finite. Under the hypotheses of Theorem 3.3,(37) lim a→∞ P (S s ℓ > s + β a ) = 0 .Proof. Define t a := s E T (a)…”

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confidence: 92%

“…An alternative, more probabilistic approach proposed in the last decade [52,53,54,40] offers some advantages regarding the above criticism. In this approach, cutoff is associated to the existence of an appropriate drift in stochastic processes or sampling schemes, and arguments are based on the behavior of hitting times [36,37,4]. Furthermore, the approach includes some general criteria for the existence of cut-off as illustrated by Proposition 3.1 below (taken from [40]).…”

Section: Introductionmentioning

confidence: 99%

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Abrupt Convergence and Escape Behavior for Birth and Death Chains

2009

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We link two phenomena concerning the asymptotical behavior of stochastic processes: (i) abrupt convergence or cut-off phenomenon, and (ii) the escape behavior usually associated to exit from metastability. The former is characterized by convergence at asymptotically deterministic times, while the convergence times for the latter are exponentially distributed. We compare and study both phenomena for discrete-time birth-and-death chains on Z with drift towards zero. In particular, this includes energy-driven evolutions with energy functions in the form of a single well. Under suitable drift hypotheses, we show that there is both an abrupt convergence towards zero and escape behavior in the other direction. Furthermore, as the evolutions are reversible, the law of the final escape trajectory coincides with the time reverse of the law of cut-off paths. Thus, for evolutions defined by

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confidence: 92%