{
  "id": "1105.5310",
  "version": "v5",
  "published": "2011-05-26T14:39:03.000Z",
  "updated": "2013-10-24T04:28:07.000Z",
  "title": "Exponentiality of first passage times of continuous time Markov chains",
  "authors": [
    "Romain Bourget",
    "Loïc Chaumont",
    "Natalia Sapoukhina"
  ],
  "categories": [
    "math.PR"
  ],
  "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.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2013-10-24T04:28:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "continuous time markov chain",
      "first passage time",
      "exponentiality",
      "quasi-stationary distribution",
      "initial law"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1105.5310B"
    }
  }
}