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Exponentiality of first passage times of continuous time Markov chains

Abstract : Let $(X,\p_x)$ be a continuous time Markov chain with finite or countable state space $S$ and let $T$ be its first passage time in a subset $D$ of $S$. It is well known that if $\mu$ is a quasi-stationary distribution relatively to $T$, then this time is exponentially distributed under $\p_\mu$. However, quasi-stationarity is not a necessary condition. In this paper, we determine more general conditions on an initial distribution $\mu$ for $T$ to be exponentially distributed under $\p_\mu$. We show in addition how quasi-stationary distributions can be expressed in terms of any initial law which makes the distribution of $T$ exponential. We also study two examples in branching processes where exponentiality does imply quasi-stationarity.
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https://hal.archives-ouvertes.fr/hal-00595912
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Submitted on : Wednesday, October 23, 2013 - 7:14:00 PM
Last modification on : Wednesday, October 20, 2021 - 3:19:10 AM
Long-term archiving on: : Friday, January 24, 2014 - 4:26:36 AM

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  • HAL Id : hal-00595912, version 5
  • ARXIV : 1105.5310

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Romain Bourget, Loïc Chaumont, Natalia Sapoukhina. Exponentiality of first passage times of continuous time Markov chains. 2012. ⟨hal-00595912v5⟩

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