Geodesic
Exact asymptotics for the~distribution of the~time of attaining the~maximum for a~trajectory of a~compound Poisson process with linear drift
Matematičeskie trudy, Tome 22 (2019) no. 2, pp. 134-156.

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

We consider the random process $at-\nu_+(pt)+\nu_-(-qt)$, $ t\in(-\infty,\infty)$, where $\nu_-$ and $\nu_+$ are independent standard Poisson processes if $t\geq 0$ and $\nu_-(t)=\nu_+(t)=0$ if $t0$. Under certain conditions on the parameters $a$, $p$, and $q$, we study the distribution function $G=G(x)$ of the time of attaining the maximum for a trajectory of this process. In the present article, we find an exact asymptotics for the tails of $G$. We also find a connection between this problem and the statistical problem of estimation of an unknown discontinuity point of a density function. 
@article{MT_2019_22_2_a7,
     author = {V. E. Mosyagin},
     title = {Exact asymptotics for the~distribution of the~time of attaining the~maximum for a~trajectory of a~compound {Poisson} process with linear drift},
     journal = {Matemati\v{c}eskie trudy},
     pages = {134--156},
     publisher = {mathdoc},
     volume = {22},
     number = {2},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2019_22_2_a7/}
}
TY  - JOUR
AU  - V. E. Mosyagin
TI  - Exact asymptotics for the~distribution of the~time of attaining the~maximum for a~trajectory of a~compound Poisson process with linear drift
JO  - Matematičeskie trudy
PY  - 2019
SP  - 134
EP  - 156
VL  - 22
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MT_2019_22_2_a7/
LA  - ru
ID  - MT_2019_22_2_a7
ER  - 
%0 Journal Article
%A V. E. Mosyagin
%T Exact asymptotics for the~distribution of the~time of attaining the~maximum for a~trajectory of a~compound Poisson process with linear drift
%J Matematičeskie trudy
%D 2019
%P 134-156
%V 22
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MT_2019_22_2_a7/
%G ru
%F MT_2019_22_2_a7
V. E. Mosyagin. Exact asymptotics for the~distribution of the~time of attaining the~maximum for a~trajectory of a~compound Poisson process with linear drift. Matematičeskie trudy, Tome 22 (2019) no. 2, pp. 134-156. http://geodesic.mathdoc.fr/item/MT_2019_22_2_a7/

[1] Borisov I. S., Mironov D. V., “Asimptoticheskoe predstavlenie otnosheniya pravdopodobiya dlya mnogomernykh vyborok s razryvnymi plotnostyami”, TVP, 45:2 (2000), 345–356 | DOI | Zbl

[2] Borisov I. S., Mironov D. V., “Asimptoticheskoe predstavlenie otnosheniya pravdopodobiya dlya neregulyarnykh semeistv raspredelenii v mnogomernom sluchae”, Sib. matem. zhurn., 42:2 (2001), 275–288 | MR | Zbl

[3] Borovkov A. A., “Predelnye teoremy o raspredelenii maksimuma summ ogranichennykh reshetchatykh sluchainykh velichin. I; II”, TVP, 5:2 (1960), 137–171 ; 5:4, 377–392 | MR | Zbl

[4] Borovkov A. A., “Novye predelnye teoremy v granichnykh zadachakh dlya summ nezavisimykh slagaemykh”, Sib. matem. zhurn., 3:5 (1962), 645–694 | Zbl

[5] Borovkov A. A., “Ob otsenivanii parametrov v sluchae razryvnykh plotnostei”, TVP, 63:2 (2018), 211–239 | DOI | MR | Zbl

[6] Ibragimov I. A., Khasminskii R. Z., Asimptoticheskaya teoriya otsenivaniya, Nauka, M., 1979 | MR

[7] Mosyagin V. E., “Otsenka skorosti skhodimosti raspredeleniya protsessa maksimalnogo pravdopodobiya v neregulyarnom sluchae”, Sib. matem. zhurn., 32:4 (1991), 96–103 | MR

[8] Mosyagin V. E., “Asimptoticheskoe predstavlenie dlya protsessa otnosheniya pravdopodobiya v sluchae razryvnoi plotnosti”, Sib. matem. zhurn., 35:2 (1994), 416–423 | MR | Zbl

[9] Mosyagin V. E., “Otsenka skorosti skhodimosti raspredelenii normirovannykh otsenok maksimalnogo pravdopodobiya v sluchae razryvnoi plotnosti”, Sib. matem. zhurn., 37:4 (1996), 895–903 | MR | Zbl

[10] Mosyagin V. E., Shvemler N. A., “Raspredelenie momenta maksimuma raznosti dvukh puassonovskikh protsessov s otritsatelnym lineinym snosom”, Sib. elektron. matem. izv., 13 (2016), 1229–1248 | MR | Zbl

[11] Mosyagin V. E., Shvemler N. A., “Lokalnye svoistva predelnogo raspredeleniya statisticheskoi otsenki tochki razryva plotnosti”, Sib. elektron. matem. izv., 14 (2017), 1307–1316 | MR | Zbl

[12] Prudnikov A. P., Brychkov Yu. A., Marichev O. I., Integraly i ryady. Elementarnye funktsii, Nauka, M., 1981 | MR

[13] Rotar V. I., Bening V. E., “Vvedenie v matematicheskuyu teoriyu strakhovaniya”, Obozrenie prikladnoi i promyshlennoi matematiki, 1:5 (1994), 698–779 | Zbl

[14] Skorokhod A. V., Sluchainye protsessy s nezavisimymi prirascheniyami, Nauka, M., 1964