Exponentiality of First Passage Times of Continuous Time Markov Chains
Let \((X,\mathbb{P}_{x})\) 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 ? is a quasi-stationary distribution relative to T, then this time is exponentially distributed under\(\mathbb {P}_{\mu}\). However, quasi-stationarity is not a necessary condition. In this paper, we determine more general conditions on an initial distribution ? for T to be exponentially distributed under\(\mathbb{P}_{\mu}\). 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.
first passage time exponential decay quasi stationary distribution 92d25 60j28