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 Electronic Journal of Probability > Vol. 11 (2006) > Paper 40 open journal systems 


Relative entropy and waiting times for continuous-time Markov processes

Jean-René Chazottes, CPht, CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, France.
Cristian Giardina, Eurandom, Postbus 513, 5600 MB Eindhoven, the Netherlands
Frank Redig, Mathematical Institute Leiden university


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).


Full text: PDF

Pages: 1049-1068

Published on: November 28, 2006


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Electronic Journal of Probability. ISSN: 1083-6489