Duffy, Ken and Macci, Claudio and Torrisi, Giovanni Luca
On the large deviations of a class of modulated additive
ESAIM: Probability and Statistics.
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 Levy 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
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