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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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Dates and versions

hal-00595912 , version 1 (25-05-2011)
hal-00595912 , version 2 (03-06-2011)
hal-00595912 , version 3 (08-07-2011)
hal-00595912 , version 4 (15-11-2012)
hal-00595912 , version 5 (23-10-2013)

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Cite

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