Exponentiality of first passage times of continuous time Markov chains

2011/05/26 by Romain Bourget, Bourget, Romain, Loïc Chaumont +3
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1105.5310

arxiv created 2013/10/24 · arxiv updated 2013/10/25

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 μ is a quasi-stationary distribution relatively to T, then this time is exponentially distributed under \p_μ. However, quasi-stationarity is not a necessary condition. In this paper, we determine more general conditions on an initial distribution μ for T to be exponentially distributed under \p_μ. 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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