On exact large deviation principles for~compound~renewal~processes
Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 2, pp. 214-230
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We consider two large deviation principles (LDPs): the “ordinary” LDP
(when the “strong” Cramér condition is met) and the “extended” LDP
when only the standard Cramér condition is met and the deviation
functional may be finite also for discontinuous trajectories. The standard
formulation of these principles involves two asymptotic (upper and lower)
estimates for the logarithms of the probabilities that the normalized
trajectory of the process lies in a given set $B$. We obtain conditions on
a set $B$ such that these estimates coincide and the large deviation
principles take the form of exact asymptotic equalities. Such LDPs are called
exact. We show that the estimating interval of an ordinary LDP is
contained in the estimating interval of the extended LDP. Hence the
fulfillment of the exact extended LDP implies that of the exact ordinary LDP.
The results obtained in the present paper are also fully valid and relevant
for random walks (a special case of compound recovery processes).
Keywords:
large deviation principle, extended large deviation principle, exact large deviation principle, most probable trajectory, deviation functional, random walks.
@article{TVP_2021_66_2_a0,
author = {A. A. Borovkov},
title = {On exact large deviation principles for~compound~renewal~processes},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {214--230},
publisher = {mathdoc},
volume = {66},
number = {2},
year = {2021},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_2021_66_2_a0/}
}
A. A. Borovkov. On exact large deviation principles for~compound~renewal~processes. Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 2, pp. 214-230. http://geodesic.mathdoc.fr/item/TVP_2021_66_2_a0/