Geodesic
A remark on normalizations in a local large deviations principle for inhomogeneous birth -- and -- death process
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1258-1269.

Voir la notice de l'article provenant de la source Math-Net.Ru

This work is a continuation of [13]. We consider a continuous-time birth – and – death process in which the transition rates are regularly varying function of the process position. We establish rough exponential asymptotic for the probability that a sample path of a normalized process lies in a neighborhood of a given nonnegative continuous function. We propose a variety of normalization schemes for which the large deviation functional preserves its natural integral form.
Keywords: birth – and – death process, normalization (scaling), large deviations principle, local large deviations principle, rate function. 
@article{SEMR_2020_17_a50,
     author = {A. V. Logachov and Y. M. Suhov and N. D. Vvedenskaya and A. A. Yambartsev},
     title = {A remark on normalizations in a local large deviations principle for inhomogeneous birth -- and -- death process},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1258--1269},
     publisher = {mathdoc},
     volume = {17},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2020_17_a50/}
}
TY  - JOUR
AU  - A. V. Logachov
AU  - Y. M. Suhov
AU  - N. D. Vvedenskaya
AU  - A. A. Yambartsev
TI  - A remark on normalizations in a local large deviations principle for inhomogeneous birth -- and -- death process
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2020
SP  - 1258
EP  - 1269
VL  - 17
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2020_17_a50/
LA  - en
ID  - SEMR_2020_17_a50
ER  - 
%0 Journal Article
%A A. V. Logachov
%A Y. M. Suhov
%A N. D. Vvedenskaya
%A A. A. Yambartsev
%T A remark on normalizations in a local large deviations principle for inhomogeneous birth -- and -- death process
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2020
%P 1258-1269
%V 17
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2020_17_a50/
%G en
%F SEMR_2020_17_a50
A. V. Logachov; Y. M. Suhov; N. D. Vvedenskaya; A. A. Yambartsev. A remark on normalizations in a local large deviations principle for inhomogeneous birth -- and -- death process. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1258-1269. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a50/

[1] A.S. Novozhilov, G.P. Karev, E.V. Koonin, “Biological applications of the theory of birth-and-death processes”, Briefings in bioinformatics, 7:1 (2006), 70–85 | DOI | MR

[2] L.M. Ricciardi, “Stochastic population theory: birth and death processes”, Biomathematics, 17 (1986), 155–190 | MR

[3] J.R. Norris, Markov chains, Reprint, Cambridge university press, Cambridge, 1998 | MR | Zbl

[4] M. Kijima, Markov processes for stochastic modeling, Chapman Hall, London, 1997 | MR | Zbl

[5] F.W. Crawford, L.S. T. Ho, M.A. Suchard, “Computational methods for birth-death processes”, WIREs Computational Statistics, 10:2 (2018) | DOI | MR

[6] A. Mogulsky, E. Pechersky, A. Yambartsev, “Large deviations for excursions of non-homogeneous Markov processes”, Electro. Commun. Probab., 19 (2014), 37, 1–8 | MR | Zbl

[7] N. Vvedenskaya, Y. Suhov, V. Belitsky, “A non-linear model of trading mechanism on a financial market”, Markov Processes. Rel. Fields, 19:1 (2013), 83–98 | MR | Zbl

[8] V.S. Korolyuk, N.I. Portenko, A.V. Skorokhod, A.F. Turbin, A manual on probability theory and mathematical statistics, Nauka, M., 1985 | MR | Zbl

[9] E.B. Dynkin, Markovskie protsessy, Springer, Berlin etc, 1965 | Zbl

[10] K. Itô, Stochastic Processes, Lecture Notes Series, 16, Matematisk Institut, Aarhus Universitet, Aarhus, 1963 | MR | Zbl

[11] A.A. Borovkov, A.A. Mogul'skii, “Iniqualities and Principle of Large Deviations for the Trajectories of Processes with Independent Increments”, Sib. Math. J., 54:2 (2013), 217–226 | DOI | MR | Zbl

[12] W. Feller., An Introduction to Probability Theory and Its Applications, v. II, Wiley, New York, 1971 | MR | Zbl

[13] N.D. Vvedenskaya, A.V. Logachov, Y.M. Suhov, A.A. Yambartsev, “Local large deviations for inhomogeneous birth-and-death processes”, Probl. Inf. Transm., 54:3 (2018), 263–280 | DOI | MR | Zbl

[14] A.V. Logachov, “The local principle of large deviations for solutions of Itô stochastic equations with quick drift”, J. Math. Sci., 218:1 (2016), 28–38 | DOI | MR | Zbl

[15] E. Seneta, Regularly Varying Sequences, Lecture Notes in Mathematics, Springer, Berlin etc., 1976 | DOI | MR | Zbl