QUASI-Stationary laws for Markov Processes: Examples of an always proximate absorbing state

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Abstract

Under consideration is a continuous-time Markov process with non-negative integer state space and a single absorbing state 0. Let T be the hitting time of zero and suppose P-i(T <x) = 1 and (*) lim(i-->x)P(i)(T > t) = 1 for all t > 0. Most known cases satisfy (*). The Markov process has a quasi-stationary distribution iff E(i)(e(epsilon T)) <infinity for some epsilon > 0.The published proof of this fact makes crucial use of (*). By means of examples it is shown that (*) can be violated in quite drastic ways without destroying the existence of a quasi-stationary distribution.
Original languageEnglish
Pages (from-to)120-145
JournalAdvances in Applied Probability
Volume27
Issue number1
DOIs
Publication statusPublished - 1995

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Quasi-stationary Distribution
Absorbing
Markov Process
Markov processes
Continuous-time Markov Process
Hitting Time
State Space
Non-negative
Integer
Zero

Cite this

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title = "QUASI-Stationary laws for Markov Processes: Examples of an always proximate absorbing state",
abstract = "Under consideration is a continuous-time Markov process with non-negative integer state space and a single absorbing state 0. Let T be the hitting time of zero and suppose P-i(T x)P(i)(T > t) = 1 for all t > 0. Most known cases satisfy (*). The Markov process has a quasi-stationary distribution iff E(i)(e(epsilon T)) 0.The published proof of this fact makes crucial use of (*). By means of examples it is shown that (*) can be violated in quite drastic ways without destroying the existence of a quasi-stationary distribution.",
author = "Anthony Pakes",
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QUASI-Stationary laws for Markov Processes: Examples of an always proximate absorbing state. / Pakes, Anthony.

In: Advances in Applied Probability, Vol. 27, No. 1, 1995, p. 120-145.

Research output: Contribution to journalArticle

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N2 - Under consideration is a continuous-time Markov process with non-negative integer state space and a single absorbing state 0. Let T be the hitting time of zero and suppose P-i(T x)P(i)(T > t) = 1 for all t > 0. Most known cases satisfy (*). The Markov process has a quasi-stationary distribution iff E(i)(e(epsilon T)) 0.The published proof of this fact makes crucial use of (*). By means of examples it is shown that (*) can be violated in quite drastic ways without destroying the existence of a quasi-stationary distribution.

AB - Under consideration is a continuous-time Markov process with non-negative integer state space and a single absorbing state 0. Let T be the hitting time of zero and suppose P-i(T x)P(i)(T > t) = 1 for all t > 0. Most known cases satisfy (*). The Markov process has a quasi-stationary distribution iff E(i)(e(epsilon T)) 0.The published proof of this fact makes crucial use of (*). By means of examples it is shown that (*) can be violated in quite drastic ways without destroying the existence of a quasi-stationary distribution.

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