Large deviations for Gaussian processes in Hölder norm

V. R. Fatalov

Izvestiya. Mathematics, Tome 67 (2003) no. 5, pp. 1061-1079

Résumé

Some results are proved on the exact asymptotic representation of large deviation probabilities for Gaussian processes in the Höder norm. The following classes of processes are considered: the Wiener process, the Brownian bridge, fractional Brownian motion, and stationary Gaussian processes with power-law covariance function. The investigation uses the method of double sums for Gaussian fields.

Bibliographie

[1] Kolmogorov A. N., “K teorii nepreryvnykh sluchainykh protsessov”, Teoriya veroyatnostei i matematicheskaya statistika, Nauka, M., 1986

[2] Loev M., Teoriya veroyatnostei, IL, M., 1962 | MR

[3] Belayev Yu. K., “Continuity and Hölder conditions for sample functions of stationary Gaussian processes”, Proc. of the Fourth Berkeley Sympos. on Math. Statistics and Probability, V. 2, 1961, 23–33 | Zbl

[4] Kramer G., Lidbetter M., Statsionarnye sluchainye protsessy, Mir, M., 1969

[5] Ciesielski Z., Kerkyacharian G., Roynette B., “Quelques espaces fonctionnels associés à des processus gaussiens”, Stud. Math., 107:2 (1993), 171–204 | MR | Zbl

[6] Fatalov V. R., “Asimptotiki veroyatnostei bolshikh uklonenii gaussovskikh polei i ikh primeneniya v teorii statistik Kolmogorova–Smirnova”, Teoriya veroyatn. i ee primen., 29:1 (1984), 178–180

[7] Fatalov V. R., “Asimptotiki veroyatnostei bolshikh uklonenii gaussovskikh polei”, Izv. AN Armenii. Matematika, 27:6 (1992), 59–81 | MR

[8] Fatalov V. R., “Asimptotiki veroyatnostei bolshikh uklonenii gaussovskikh polei. Primeneniya”, Izv. AN Armenii. Matematika, 28:5 (1993), 32–55 | MR

[9] Piterbarg V. I., Asimptoticheskie metody v teorii gaussovskikh sluchainykh protsessov i polei, Izd-vo MGU, M., 1988

[10] Piterbarg V. I., Fatalov V. R., “Metod Laplasa dlya veroyatnostnykh mer v banakhovykh prostranstvakh”, UMN, 50:6 (1995), 57–150 | MR | Zbl

[11] Baldi P., Roynette B., “Some exact equivalents for Brownian motion in Hölder norm”, Prob. Theory Rel. Fields, 93:4 (1992), 457–484 | DOI | MR | Zbl

[12] Baldi P., Ben Arous G., Kerkyacharian G., “Large deviations and the Strassen theorem in Hölder norm”, Stoch. Proc. Appl., 42 (1992), 171–180 | DOI | MR | Zbl

[13] Lorang G., Roynette B., “Un théorème de Schilder pour des fonctionnelles browniennes non régulières”, Ann. Inst. H. Poincaré. Probab. et Statist., 29:4 (1993), 513–530 | MR | Zbl

[14] Kolmogorov A. N., “Spiral Vinera i nekotorye drugie interesnye krivye v gilbertovom prostranstve”, DAN SSSR, 26:2 (1940), 115–118 | MR | Zbl

[15] Mandelbrot B. B., The fractional geometry of nature, Freeman, San Francisco, 1982 | MR | Zbl

[16] Michna Z., “On tail probabilities and first passage times for fractional Brownian motion”, Math. Methods Oper. Res., 49:2 (1999), 335–354 | MR | Zbl

[17] Pickands J. III., “Upcrossing probabilities for stationary Gaussian processes”, Trans. Amer. Math. Soc., 145 (1969), 51–73 | DOI | MR | Zbl

[18] Lidbetter M., Lindgren G., Rotsen Kh., Ekstremumy sluchainykh posledovatelnostei i protsessov, Mir, M., 1989 | MR

[19] Shao Q.-M., “Bounds and estimators of a basic constant in extreme value theory of Gaussian processes”, Statist. Sinica, 6 (1996), 245–257 | MR | Zbl

[20] Yakir B., Pollak M., “A new representation for a renewal–theoretic constant appearing in asymptotic approximations of large deviations”, Ann. Appl. Probab., 8:3 (1998) | MR | Zbl

[21] Fatalov V. R., “Asimptotiki veroyatnostei malykh uklonenii v $L^2$-norme dlya dvukh klassov gaussovskikh statsionarnykh protsessov”, Teoriya veroyatn. i ee primen. (to appear)

[22] Fatalov V. R., “Konstanty v asimptotikakh veroyatnostei malykh uklonenii dlya gaussovskikh protsessov i polei”, UMN, 58:4 (2003), 89–134 | MR | Zbl

[23] Adler R. J., “On excursion sets, tube formulas and maxima of random fields”, Ann. Appl. Prob., 10:1 (2000), 1–74 | MR | Zbl

[24] Piterbarg V. I., Prisyazhnyuk V. P., “Tochnaya asimptotika veroyatnosti bolshogo razmakha gaussovskogo statsionarnogo protsessa”, Teoriya veroyatn. i ee primen., 26:3 (1981), 480–495 | MR | Zbl

[25] Lifshits M. A., Gaussovskie sluchainye funktsii, TViMS, Kiev, 1995 | MR | Zbl

[26] Borodin A. N., Salminen P., Spravochnik po brounovskomu dvizheniyu, Lan, S.-Pb., 2000 | Zbl

[27] Ben Arous G., Ledoux M., “Grandes déviations de Freidlin–Wentzell en norme hölderienne”, Lect. Notes Math., 1583, 1994, 293–299 | MR | Zbl

[28] Ledoux M., “Isoperimetry and Gaussian analysis”, Lect. Notes Math., 1648, 1996, 165–294 | MR | Zbl

[29] N'zi M., “Strassen theorem in Hölder norm for some Brownian functionals”, Ann. Math. Blaise Pascal, 4:2 (1997), 49–59 | MR | Zbl

[30] Fatalov V. R., “Bolshie ukloneniya gaussovskikh mer v prostranstvakh $l^p$ i $L^p$, $p\geqslant 2$”, Teoriya veroyatn. i ee primen., 41:3 (1996), 682–689 | MR | Zbl

[31] Fatalov V. R., “Asimptotiki bolshikh uklonenii vinerovskikh polei v $L^p$-norme, nelineinye uravneniya Khammershteina i giperbolicheskie kraevye zadachi vysokogo poryadka”, Teoriya veroyatn. i ee primen., 47:4 (2002), 711–726 | MR

[32] Fatalov V. R., “Asimptotiki bolshikh uklonenii gaussovskikh protsessov tipa vinerovskogo dlya $L^p$-funktsionalov, $p>0$, i gipergeometricheskaya funktsiya”, Matem. sb., 194:3 (2003), 61–82 | MR | Zbl