Extreme value statistics of ergodic Markov processes from first passage times in the large deviation limit

Using a matrix approach we discuss the first-passage time of a Markov process to exceed a given threshold or for the maximal increment of this process to pass a certain critical value. Conditions under which this first-passage time possesses various ageing properties are studied. Some results previously obtained by Li and Shaked (1995) are extended.

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The first-passage times of phase semi-Markov processes

Statistics & Probability Letters ◽

10.1016/j.spl.2011.08.021 ◽

2012 ◽

Vol 82 (1) ◽

pp. 40-48 ◽

Cited By ~ 4

Author(s):

Xuan Zhang ◽

Zhenting Hou

Keyword(s):

Markov Processes ◽

First Passage Times ◽

First Passage ◽

Semi Markov Processes ◽

Passage Times

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First passage times and their algorithms for markov processes

Communications in Statistics Stochastic Models ◽

10.1080/15326349508807338 ◽

1995 ◽

Vol 11 (1) ◽

pp. 195-210 ◽

Cited By ~ 9

Author(s):

HSU Guang-Hui ◽

Yuan Xue-Ming

Keyword(s):

Markov Processes ◽

First Passage Times ◽

First Passage ◽

Passage Times

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On ageing properties of first-passage times of increasing Markov processes

Advances in Applied Probability ◽

10.1017/s0001867800011484 ◽

2002 ◽

Vol 34 (01) ◽

pp. 241-259

Author(s):

Félix Belzunce ◽

Eva-María Ortega ◽

José M. Ruiz

Keyword(s):

Markov Chain ◽

Markov Processes ◽

Continuous Time ◽

Transition Matrix ◽

Sufficient Conditions ◽

First Passage Times ◽

Generator Matrix ◽

First Passage ◽

Finite Case ◽

Passage Times

The purpose of this paper is to study ageing properties of first-passage times of increasing Markov chains. We extend the literature to some new ageing classes, such as the IFR(2), NBU(2), DRLLt and NBULt classes. We also give sufficient conditions in the finite case, that are more efficient computationally, just in terms of the transition matrix K, in the discrete case, or the generator matrix Q, in the continuous case. For the uniformizable, continuous-time Markov processes, we derive conditions in terms of the discrete uniformized Markov chain for the NBU(2) and the NBULt classes. In the last section, a review of the main results in this direction in the literature is given, and we compare some of the conditions stated in this paper with others given in the literature about some other ageing classes. Some examples where these results are applied are given.

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