We prove that the large deviation principle holds for a class of processes inspired by semi-Markov additive processes. For the processes we consider, the sojourn times in the phase process need not be independent and identically distributed. Moreover the state selection process need not be independent of the sojourn times. We assume that the phase process takes values in a finite set and that the order in which elements in the set, called states, are visited is selected stochastically. The sojourn times determine how long the phase process spends in a state once it has been selected. The main tool is a representation formula for the sample paths of the empirical laws of the phase process. Then, based on assumed joint large deviation behavior of the state selection and sojourn processes, we prove that the empirical laws of the phase process satisfy a sample path large deviation principle. From this large deviation principle, the large deviations behavior of a class of modulated additive processes is deduced. As an illustration of the utility of the general results, we provide an alternate proof of results for modulated Lévy processes. As a practical application of the results, we calculate the large deviation rate function for a processes that arises as the International Telecommunications Union's standardized stochastic model of two-way conversational speech.
Mots-clés : large deviations, modulated additive processes, speech models
@article{PS_2011__15__83_0,
author = {Duffy, Ken R. and Macci, Claudio and Torrisi, Giovanni Luca},
title = {On the large deviations of a class of modulated additive processes},
journal = {ESAIM: Probability and Statistics},
pages = {83--109},
publisher = {EDP-Sciences},
volume = {15},
year = {2011},
doi = {10.1051/ps/2009010},
mrnumber = {2870507},
zbl = {1267.60030},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2009010/}
}
TY - JOUR AU - Duffy, Ken R. AU - Macci, Claudio AU - Torrisi, Giovanni Luca TI - On the large deviations of a class of modulated additive processes JO - ESAIM: Probability and Statistics PY - 2011 SP - 83 EP - 109 VL - 15 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2009010/ DO - 10.1051/ps/2009010 LA - en ID - PS_2011__15__83_0 ER -
%0 Journal Article %A Duffy, Ken R. %A Macci, Claudio %A Torrisi, Giovanni Luca %T On the large deviations of a class of modulated additive processes %J ESAIM: Probability and Statistics %D 2011 %P 83-109 %V 15 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2009010/ %R 10.1051/ps/2009010 %G en %F PS_2011__15__83_0
Duffy, Ken R.; Macci, Claudio; Torrisi, Giovanni Luca. On the large deviations of a class of modulated additive processes. ESAIM: Probability and Statistics, Tome 15 (2011), pp. 83-109. doi : 10.1051/ps/2009010. http://www.numdam.org/articles/10.1051/ps/2009010/
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