Reversal of Markov Chains and the Forget Time

L. Lovász, Peter Winkler

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

We study three quantities that can each be viewed as the time needed for a finite irreducible Markov chain to 'forget' where it started. One of these is the mixing time, the minimum mean length of a stopping rule that yields the stationary distribution from the worst starting state. A second is the forget time, the minimum mean length of any stopping rule that yields the same distribution from any starting state. The third is the reset time, the minimum expected time between independent samples from the stationary distribution. Our main results state that the mixing time of a chain is equal to the mixing time of the time-reversed chain, while the forget time of a chain is equal to the reset time of the reverse chain. In particular, the forget time and the reset time of a time-reversible chain are equal. Moreover, the mixing time lies between absolute constant multiples of the sum of the forget time and the reset time. We also derive an explicit formula for the forget time, in terms of the 'access times' introduced in [11]. This enables us to relate the forget and reset times to other mixing measures of the chain.

Original languageEnglish
Pages (from-to)189-204
Number of pages16
JournalCombinatorics Probability and Computing
Volume7
Issue number2
Publication statusPublished - 1998

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Reversal
Markov processes
Markov chain
Mixing Time
Stopping Rule
Stationary Distribution
Reverse
Explicit Formula

ASJC Scopus subject areas

  • Computational Theory and Mathematics
  • Mathematics(all)
  • Discrete Mathematics and Combinatorics
  • Statistics and Probability
  • Theoretical Computer Science

Cite this

Reversal of Markov Chains and the Forget Time. / Lovász, L.; Winkler, Peter.

In: Combinatorics Probability and Computing, Vol. 7, No. 2, 1998, p. 189-204.

Research output: Contribution to journalArticle

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