Geodesic
On the distribution of the first passage time of an arbitrary remote boundary by random walk
Teoriâ veroâtnostej i ee primeneniâ, Tome 61 (2016) no. 2, pp. 210-233 
A. A. Borovkov. On the distribution of the first passage time of an arbitrary remote boundary by random walk. Teoriâ veroâtnostej i ee primeneniâ, Tome 61 (2016) no. 2, pp. 210-233. http://geodesic.mathdoc.fr/item/TVP_2016_61_2_a1/
@article{TVP_2016_61_2_a1,
     author = {A. A. Borovkov},
     title = {On the distribution of the first passage time of an arbitrary remote boundary by random walk},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {210--233},
     year = {2016},
     volume = {61},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2016_61_2_a1/}
}
TY  - JOUR
AU  - A. A. Borovkov
TI  - On the distribution of the first passage time of an arbitrary remote boundary by random walk
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2016
SP  - 210
EP  - 233
VL  - 61
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TVP_2016_61_2_a1/
LA  - ru
ID  - TVP_2016_61_2_a1
ER  - 
%0 Journal Article
%A A. A. Borovkov
%T On the distribution of the first passage time of an arbitrary remote boundary by random walk
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2016
%P 210-233
%V 61
%N 2
%U http://geodesic.mathdoc.fr/item/TVP_2016_61_2_a1/
%G ru
%F TVP_2016_61_2_a1

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

Under conditions close to the minimal conditions, we find the local and integral asymptotics for the joint distribution of the first passage time by a random walk of an arbitrary remote boundary and the overshoot over that boundary.

[1] Borovkov A. A., Teoriya veroyatnostei, Editorial URSS, M., 2009, 652 pp.

[2] Anscombe F. J., “Large-sample theory of sequential estimation”, Proc. Cambridge Philos. Soc., 48 (1952), 600–607 | DOI | MR | Zbl

[3] Siegmund D., “On the asymptotic normality of one-sided stopping rules”, Ann. Math. Statist., 39:5 (1968), 1493–1497 | MR | Zbl

[4] Gut A., “On the moments and limit distributions of some first passage times”, Ann. Probab., 2:2 (1974), 277–308 | DOI | MR | Zbl

[5] Lalley S. P., “Limit theorems for first-passage times in linear and non-linear renewal theory”, Adv. Appl. Probab., 16:4 (1984), 766–803 | DOI | MR | Zbl

[6] Woodroofe M., Keener R., “Asymptotic expansions in boundary crossing problems”, Ann. Probab., 15:1 (1987), 102–114 | DOI | MR | Zbl

[7] Ragimov F. G., “O lokalnykh veroyatnostyakh peresecheniya nelineinykh granits summami nezavisimykh sluchainykh velichin”, Teoriya veroyatn. i ee primen., 35:2 (1990), 373–376 | MR

[8] Ragimov F. G., “Asimptoticheskoe razlozhenie raspredeleniya vremeni peresecheniya nelineinykh granits”, Teoriya veroyatn. i ee primen., 37:3 (1992), 580–583 | MR | Zbl

[9] Ragimov F. G., “Integralnye predelnye teoremy dlya vremeni peresecheniya nelineinykh granits summami nezavisimykh velichin”, Teoriya veroyatn. i ee primen., 50:1 (2005), 158–161 | DOI

[10] Aliev S. A., Hashimova T. E., “Asymptotic behaviour of the conditional probability of the nonlinear boundary crossing by a random walk”, Theory Stoch. Process., 16:1 (2010), 12–16 | MR | Zbl

[11] Borovkov A. A., Borovkov K. A., Asimptoticheskii analiz sluchainykh bluzhdanii. Medlenno ubyvayuschie raspredeleniya skachkov, Fizmatlit, M., 2008, 650 pp.

[12] Borovkov A. A., Asimptoticheskii analiz sluchainykh bluzhdanii. Bystro ubyvayuschie raspredeleniya priraschenii, Fizmatlit, M., 2013, 447 pp.

[13] Stone C., “A local limit theorem for nonlattice multidimensional distribution functions”, Ann. Math. Statistics, 36 (1965), 546–551 | DOI | MR | Zbl

[14] Stone C., “On local and ratio limit theorems”, Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, part II, v. II, Univ.of California, Berkeley–Los Angeles, 1967, 217–224 | MR | Zbl

[15] Woodroofe M., Nonlinear Renewal Theory in Sequential Analysis, SIAM, Providence, 1982, 119 pp. | MR