Geodesic
Method of moments for exit probabilities of Gaussian vector processes from a large region
Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 4, pp. 669-682 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Asymptotic behavior of the exit probability from homothetically unboundedly expanding region is evaluated for twice continuously differentiable stationary Gaussian vector processes. The evaluation is based on the Rice's method, that is, on studying point processes of exit times from the region, with subsequent application of the Laplace asymptotic method.
Keywords: Gaussian vector process, large excursions of trajectories, asymptotic behavior, Rice's method, Laplace method, point process. 
@article{TVP_2018_63_4_a2,
     author = {A. O. Kleban and V. I. Piterbarg},
     title = {Method of moments for exit probabilities of {Gaussian} vector processes from a large region},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {669--682},
     year = {2018},
     volume = {63},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2018_63_4_a2/}
}
TY  - JOUR
AU  - A. O. Kleban
AU  - V. I. Piterbarg
TI  - Method of moments for exit probabilities of Gaussian vector processes from a large region
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2018
SP  - 669
EP  - 682
VL  - 63
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/TVP_2018_63_4_a2/
LA  - ru
ID  - TVP_2018_63_4_a2
ER  - 
%0 Journal Article
%A A. O. Kleban
%A V. I. Piterbarg
%T Method of moments for exit probabilities of Gaussian vector processes from a large region
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2018
%P 669-682
%V 63
%N 4
%U http://geodesic.mathdoc.fr/item/TVP_2018_63_4_a2/
%G ru
%F TVP_2018_63_4_a2
A. O. Kleban; V. I. Piterbarg. Method of moments for exit probabilities of Gaussian vector processes from a large region. Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 4, pp. 669-682. http://geodesic.mathdoc.fr/item/TVP_2018_63_4_a2/

[1] J.-M. Azaïs, M. Wschebor, Level sets and extrema of random processes and fields, John Wiley Sons, Inc., Hoboken, NJ, 2009, xii+393 pp. | DOI | MR | Zbl

[2] K. Debicki, K. M. Kosiński, M. Mandjes, T. Rolski, “Extremes of multidimensional Gaussian processes”, Stochastic Process. Appl., 120:12 (2010), 2289–2301 | DOI | MR | Zbl

[3] J. Farkas, E. Hashorva, V. I. Piterbarg, “Asymptotic behavior of reliability function for multidimensional aggregated Weibull type reliability indices”, Analytical and computational methods in probability theory (Moscow, 2017), Lecture Notes in Comput. Sci., 10684, Springer, Cham, 2017, 251–264 | DOI | Zbl

[4] F. Avram, Z. Palmowski, M. Pistorius, “A two-dimensional ruin problem on the positive quadrant”, Insurance Math. Econom., 42:1 (2008), 227–234 | DOI | MR | Zbl

[5] E. Hashorva, D. Korshunov, V. I. Piterbarg, “Asymptotic expansion of Gaussian chaos via probabilistic approach”, Extremes, 18:3 (2015), 315–347 | DOI | MR | Zbl

[6] O. Kallenberg, Random measures, Schriftenreihe des Zentralinstituts fur Mathematik und Mechanik bei der Akademie der Wissenschaften der DDR, 23, Akademie-Verlag, Berlin, 1975, vi+104 pp. | MR | Zbl

[7] D. A. Korshunov, V. I. Piterbarg, E. Hashorva, “On extremal behavior of Gaussian chaos”, Dokl. Math., 88:2 (2013), 566–568 | DOI | DOI | MR | Zbl

[8] R. Illsley, “The moments of the number of exits from a simply connected region”, Adv. in Appl. Probab., 30:1 (1998), 167–180 | DOI | MR | Zbl

[9] D. A. Korshunov, V. I. Piterbarg, E. Hashorva, “On the asymptotic Laplace method and its application to random chaos”, Math. Notes, 97:6 (2015), 878–891 | DOI | DOI | MR | Zbl

[10] V. I. Piterbarg, “High extrema of Gaussian chaos processes”, Extremes, 19:2 (2016), 253–272 | DOI | MR | Zbl

[11] A. I. Zhdanov, V. I. Piterbarg, “Bolshie vybrosy protsessov gaussovskogo khaosa. Approksimatsiya v diskretnom vremeni”, Teoriya veroyatn. i ee primen., 63:1 (2018), 3–28 | DOI | MR

[12] V. Piterbarg, S. Stamatović, “Limit theorem for high level $a$-upcrossings by $\chi$-process”, Theory Probab. Appl., 48:4 (2004), 734–741 | DOI | DOI | MR | Zbl

[13] A. P. Trifonov, Yu. S. Shinakov, Sovmestnoe razlichenie signalov i otsenka ikh parametrov na fone pomekh, Statisticheskaya teoriya svyazi, 26, Radio i svyaz, M., 1986, 264 pp. | MR

[14] V. I. Piterbarg, Asymptotic methods in the theory of Gaussian processes and fields, Transl. Math. Monogr., 148, reprint ed., Amer. Math. Soc., Providence, RI, 2012, xii+206 pp. | DOI | MR | Zbl | Zbl

[15] V. I. Piterbarg, “Metod Raisa dlya gaussovskikh sluchainykh polei”, Fundament. i prikl. matem., 2:1 (1996), 187–204 | MR | Zbl

[16] V. I. Piterbarg, Twenty lectures about Gaussian processes, Atlantic Financial Press, London, 2015, xi+167 pp. | Zbl

[17] S. O. Rice, “Mathematical analysis of random noise”, Bell System Tech. J., 23:3 (1944), 282–332 ; 24:1 (1945), 46–156 | DOI | MR | Zbl | DOI | MR | Zbl

[18] V. I. Piterbarg, “Massivnye vybrosy gladkikh gaussovskikh izotropnykh polei. Metod momentov”, Teoriya veroyatn. i ee primen., 63:2 (2018), 240–259 | DOI | MR

[19] E. V. Kremena, V. I. Piterbarg, J. Hüsler, “On the shape of trajectories of Gaussian processes having large massive excursions”, Theory Probab. Appl., 58:4 (2014), 582–600 | DOI | DOI | MR | Zbl

[20] J. Hüsler, E. V. Kremena, V. I. Piterbarg, “On the shape of trajectories of Gaussian processes having large massive excursions. II”, Theory Probab. Appl., 60:3 (2016), 513–520 | DOI | DOI | MR | Zbl

[21] J. Wojdylo, “Computing the coefficients in Laplace's method”, SIAM Rev., 48:1 (2006), 76–96 | DOI | MR | Zbl