On the large deviations of a class of modulated additive processes

Ken R. Duffy; Claudio Macci; Giovanni Luca Torrisi

ESAIM: Probability and Statistics (2012)

  • Volume: 15, page 83-109
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Duffy, Ken R., Macci, Claudio, and Torrisi, Giovanni Luca. "On the large deviations of a class of modulated additive processes." ESAIM: Probability and Statistics 15 (2012): 83-109. <http://eudml.org/doc/222463>.

@article{Duffy2012,
abstract = { 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. },
author = {Duffy, Ken R., Macci, Claudio, Torrisi, Giovanni Luca},
journal = {ESAIM: Probability and Statistics},
keywords = {Large deviations; modulated additive processes; speech models; large deviation; modulated additive process; speech model},
language = {eng},
month = {1},
pages = {83-109},
publisher = {EDP Sciences},
title = {On the large deviations of a class of modulated additive processes},
url = {http://eudml.org/doc/222463},
volume = {15},
year = {2012},
}

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
DA - 2012/1//
PB - EDP Sciences
VL - 15
SP - 83
EP - 109
AB - 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.
LA - eng
KW - Large deviations; modulated additive processes; speech models; large deviation; modulated additive process; speech model
UR - http://eudml.org/doc/222463
ER -

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