# Exponentiality of first passage times of continuous time Markov chains

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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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Cited literature [29 references]

https://hal.archives-ouvertes.fr/hal-00595912
Contributor : Loïc Chaumont <>
Submitted on : Wednesday, October 23, 2013 - 7:14:00 PM
Last modification on : Tuesday, March 17, 2020 - 2:14:23 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

### Citation

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

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