Geodesic
Asymptotics for the distribution of the time of attaining the maximum for a trajectory of a Poisson process with linear drift and intensity switch
Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 1, pp. 94-109

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

We find the exact asymptotics of the distribution of the time when the trajectory of the process $Y(t)=at-\nu_+(pt)+\nu_-(-qt)$, $t\in(-\infty,\infty)$ attains its maximum, where $\nu_{\pm}(t)$ are independent standard Poisson processes extended by zero on the negative semiaxis. The parameters $a$, $p$, $q$ are assumed just to satisfy the condition $\mathbf{E}Y(t)<0$, $t\neq 0$.
Keywords: Poisson process with linear drift, random process with negative mean drift, exact asymptotics of distribution tails. 
V. E. Mosyagin. Asymptotics for the distribution of the time of attaining the maximum for a trajectory of a Poisson process with linear drift and intensity switch. Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 1, pp. 94-109. http://geodesic.mathdoc.fr/item/TVP_2021_66_1_a4/
@article{TVP_2021_66_1_a4,
     author = {V. E. Mosyagin},
     title = {Asymptotics for the distribution of the time of attaining the maximum for a~trajectory of {a~Poisson} process with linear drift and intensity switch},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {94--109},
     year = {2021},
     volume = {66},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2021_66_1_a4/}
}
TY  - JOUR
AU  - V. E. Mosyagin
TI  - Asymptotics for the distribution of the time of attaining the maximum for a trajectory of a Poisson process with linear drift and intensity switch
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2021
SP  - 94
EP  - 109
VL  - 66
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TVP_2021_66_1_a4/
LA  - ru
ID  - TVP_2021_66_1_a4
ER  - 
%0 Journal Article
%A V. E. Mosyagin
%T Asymptotics for the distribution of the time of attaining the maximum for a trajectory of a Poisson process with linear drift and intensity switch
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2021
%P 94-109
%V 66
%N 1
%U http://geodesic.mathdoc.fr/item/TVP_2021_66_1_a4/
%G ru
%F TVP_2021_66_1_a4

[1] A. V. Skorokhod, Random processes with independent increments, Math. Appl. (Soviet Ser.), 47, Kluwer Acad. Publ., Dordrecht, 1991, xii+279 pp. | MR | MR | Zbl | Zbl

[2] A. A. Borovkov, “On estimation of parameters in the case of discontinuous densities”, Theory Probab. Appl., 63:2 (2018), 169–192 | DOI | DOI | MR | Zbl

[3] I. A. Ibragimov, R. Z. Has'minskiĭ, Statistical estimation. Asymptotic theory, Appl. Math., 16, Springer-Verlag, New York–Berlin, 1981, vii+403 pp. | MR | MR | Zbl | Zbl

[4] V. E. Mosyagin, “An estimate of the rate of convergence of the distribution of the maximum probability process in the nonregular case”, Siberian Math. J., 32:4 (1991), 616–622 | DOI | MR | Zbl

[5] V. E. Mosyagin, “Asymptotic representation of the process of the likelihood ratio in the case of a discontinuous density”, Siberian Math. J., 35:2 (1994), 375–382 | DOI | MR | Zbl

[6] V. E. Mosyagin, “Estimation of the convergence rate for the distributions of normalized maximum likelihood estimators in the case of a discontinuous density”, Siberian Math. J., 37:4 (1996), 788–796 | DOI | MR | Zbl

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

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

[9] I. S. Borisov, D. V. Mironov, “An asymptotic representation for the likelihood ratio for multidimensional samples with discontinuous desities”, Theory Probab. Appl., 45:2 (2001), 289–300 | DOI | DOI | MR | Zbl

[10] I. S. Borisov, D. V. Mironov, “An asymptotic representation of the likelihood ratio for irregular families of distributions in the multivariate case”, Siberian Math. J., 42:2 (2001), 232–244 | DOI | MR | Zbl