### Abstract

For discrete-time stochastic processes, there is a close connection between return (resp. waiting) times and entropy (resp. relative entropy). Such a connection cannot be straightforwardly extended to the continuous-time setting. Contrarily to the discrete-time case one needs a reference measure on path space and so the natural object is relative entropy rather than entropy. In this paper we elaborate on this in the case of continuous-time Markov processes with finite state space. A reference measure of special interest is the one associated to the time-reversed process. In that case relative entropy is interpreted as the entropy production rate. The main results of this paper are: almost-sure convergence to relative entropy of the logarithm of waiting-times ratios suitably normalized, and their fluctuation properties (central limit theorem and large deviation principle)

Original language | English |
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Pages (from-to) | 1049-1068 |

Journal | Electronic Journal of Probability |

Volume | 11 |

Publication status | Published - 2006 |

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## Cite this

Chazottes, J. R., Giardinà, C., & Redig, F. H. J. (2006). Relative entropy and waiting time for continuous-time Markov processes.

*Electronic Journal of Probability*,*11*, 1049-1068.