## Survival in a quasi-death process (2007)

Citations: | 2 - 1 self |

### BibTeX

@MISC{Doorn07survivalin,

author = {Erik A. Van Doorn and Phil Pollett},

title = {Survival in a quasi-death process},

year = {2007}

}

### OpenURL

### Abstract

Abstract. We consider a Markov chain in continuous time with one absorbing state and a finite set S of transient states. When S is irreducible the limiting distribution of the chain as t → ∞, conditional on survival up to time t, is known to equal the (unique) quasi-stationary distribution of the chain. We address the problem of generalizing this result to a setting in which S may be reducible, and obtain a complete solution if the eigenvalue with maximal real part of the generator of the (sub)Markov chain on S has geometric (but not, necessarily, algebraic) multiplicity one. The result is applied to pure death processes and, more generally, to quasi-death processes.

### Citations

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(Show Context)
Citation Context ... t) = lim t→∞ as required. ∑ i∈S wi ∑ i∈S wi ∑ j∈S Pij(t + s) ∑ j∈S Pij(t) = e−αs , Remarks (i) The results in [14] and [5] constitute the continuous-time counterparts of results obtained in [13] and =-=[4]-=-, respectively, in a discrete-time setting. The latter results have been generalized (in a more abstract, but still discrete, setting) by Lindqvist [12]. An alternative approach towards proving Theore... |

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Citation Context ...nteraction of the initial distribution and the distribution over S known as the quasi-stationary distribution of the process. Similar ideas have been put forward independently by Steinsaltz and Evans =-=[17]-=-. Aalen and Gjessing discuss several examples of relevant stochastic processes, including finite-state Markov chains with an absorbing state, which is the setting of the present paper. A survival-time... |

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Citation Context ...h process, limiting conditional distribution, quasi-stationary distribution, survival-time distribution 2000 Mathematics Subject Classification: Primary 60J271 Introduction In the interesting papers =-=[2]-=- and [3] Aalen and Gjessing provide a new explanation for the shape of hazard rate functions in survival analysis. They propose to model survival times as sojourn times of stochastic processes in a se... |

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Citation Context ...ng state, which is the setting of the present paper. A survival-time distribution in this setting is known as a phase-type distribution (see, for example, Latouche and Ramaswami [10, Ch. 2], or Aalen =-=[1]-=-). In their analysis and examples Aalen and Gjessing restrict themselves to chains for which the set S of transient states constitutes a single class, arguing that “irreducibility is important when co... |

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Citation Context ...+ s | T > t) = lim t→∞ as required. ∑ i∈S wi ∑ i∈S wi ∑ j∈S Pij(t + s) ∑ j∈S Pij(t) = e−αs , Remarks (i) The results in [14] and [5] constitute the continuous-time counterparts of results obtained in =-=[13]-=- and [4], respectively, in a discrete-time setting. The latter results have been generalized (in a more abstract, but still discrete, setting) by Lindqvist [12]. An alternative approach towards provin... |

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Citation Context ...ime counterparts of results obtained in [13] and [4], respectively, in a discrete-time setting. The latter results have been generalized (in a more abstract, but still discrete, setting) by Lindqvist =-=[12]-=-. An alternative approach towards proving Theorem 5 would be to take Lindqvist results (in particular [12, Theorem 5.8]) as a starting point and prove their analogues in a continuous setting. (ii) The... |

2 |
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(Show Context)
Citation Context ...s, limiting conditional distribution, quasi-stationary distribution, survival-time distribution 2000 Mathematics Subject Classification: Primary 60J271 Introduction In the interesting papers [2] and =-=[3]-=- Aalen and Gjessing provide a new explanation for the shape of hazard rate functions in survival analysis. They propose to model survival times as sojourn times of stochastic processes in a set S of t... |

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Citation Context ... analogues in a continuous setting. (ii) The fact that the limiting distribution of the residual survival time exists and is exponentially distributed has been observed by Kalpakam [8] and Li and Cao =-=[11]-=- in a somewhat more general setting, namely when the Laplace transform of the survival-time distribution is a rational function (cf. [15]). ✷ In what follows we are interested in particular in propert... |

1 |
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Citation Context ...int and prove their analogues in a continuous setting. (ii) The fact that the limiting distribution of the residual survival time exists and is exponentially distributed has been observed by Kalpakam =-=[8]-=- and Li and Cao [11] in a somewhat more general setting, namely when the Laplace transform of the survival-time distribution is a rational function (cf. [15]). ✷ In what follows we are interested in p... |

1 |
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(Show Context)
Citation Context ...y distributed has been observed by Kalpakam [8] and Li and Cao [11] in a somewhat more general setting, namely when the Laplace transform of the survival-time distribution is a rational function (cf. =-=[15]-=-). ✷ In what follows we are interested in particular in properties of the left eigenvector u that are determined by structural properties of Q. To set the stage we first look more closely into the sim... |