On the Laplace method for Gaussian measures in a Banach space

V. R. Fatalov

Teoriâ veroâtnostej i ee primeneniâ, Tome 58 (2013) no. 2, pp. 325-354

Résumé

In this paper, we prove results on sharp asymptotics for the probabilities $P_A(uD)$, as $u\to\infty$, where $P_A$ is the Gaussian measure in an infinite-dimensional Banach space $B$ with zero mean and nondegenerate covariance operator $A$, $D=\{x\in B:Q(x)\geqslant 0\}$ is a Borel set in $B$, and $Q$ is a smooth function. We analyze the case where the action functional attains its minimum on some set $D$ on a one-dimensional manifold. We make use of the Laplace method in Banach spaces for Gaussian measures. Based on the general result obtained, for $0<p\leqslant6$ we find a sharp asymptotics for large deviations of distributions of $L^p$-functionals for the centered Brownian bridge which arises as the limit while studying the Watson statistics. Explicit constants are given for the cases $p=1$ and $p=2$.

Détail

Keywords: Laplace’s method; large deviations; gaussian process; principle of large deviations; action functional; centered Brownian bridge; Watson statistics; hypergeometric function.

Bibliographie

[1] Abramovits M., Stigan I. (red.), Spravochnik po spetsialnym funktsiyam, Nauka, M., 1979, 830 pp.

[2] Alekseev V. M., Tikhomirov V. M., Fomin S. V., Optimalnoe upravlenie, Nauka, M., 1978, 429 pp.

[3] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii: Gipergeometricheskaya funktsiya. Funktsii Lezhandra, Nauka, M., 1973, 294 pp.

[4] Borovkov A. A., Mogulskii A. A., “O veroyatnostyakh bolshikh uklonenii v topologicheskikh prostranstvakh. I; II”, Sibir. matem. zhurn., 19:5 (1978), 988–1004 ; 21:5 (1980), 12–26 | MR | Zbl | MR | Zbl

[5] Vainberg M. M., Variatsionnyi metod i metod monotonnykh operatorov v teorii nelineinykh operatorov, Nauka, M., 1972, 416 pp.

[6] Vakhaniya N. N., Tarieladze V. I., Chobanyan S. A., Veroyatnostnye raspredeleniya v banakhovykh prostranstvakh, Nauka, M., 1985, 369 pp.

[7] Venttsel A. D., Freidlin M. I., Fluktuatsii v dinamicheskikh sistemakh pod deistviem malykh sluchainykh vozmuschenii, Nauka, M., 1979, 424 pp.

[8] Venttsel A. D., Predelnye teoremy o bolshikh ukloneniyakh dlya markovskikh sluchainykh protsessov, Nauka, M., 1986, 175 pp.

[9] Gikhman I. I., Skorokhod A. V., Teoriya sluchainykh protsessov, v. 1, Nauka, M., 1971, 664 pp.

[10] Go Kh.-S., Gaussovskie mery v banakhovykh prostranstvakh, Nauka, M., 1979, 176 pp.

[11] Gokhberg I. Ts., Krein M. G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, M., 1965, 448 pp.

[12] Danford N., Shvarts Dzh. T., Lineinye operatory. Obschaya teoriya, URSS, M., 2004, 895 pp.

[13] Zolotarev V. M., “Ob odnoi veroyatnostnoi zadache”, Teoriya veroyatn. i ee primen., 6:2 (1961), 219–222 | MR

[14] Kamke E., Spravochnik po obyknovennym differentsialnym uravneniyam, Nauka, M., 1965, 703 pp.

[15] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz, Nauka, M., 1977, 741 pp.

[16] Koddington E. A., Levinson N., Teoriya obyknovennykh differentsialnykh uravnenii, URSS, M., 2007, 474 pp.

[17] Krasnoselskii M. A., Zabreiko P. P., Pustylnik E. I., Sobolevskii P. E., Integralnye operatory v prostranstvakh summiruemykh funktsii, Nauka, M., 1966, 499 pp.

[18] Krein S. G. (red.), Funktsionalnyi analiz, Nauka, M., 1972, 544 pp.

[19] Lankaster P., Teoriya matrits, Nauka, M., 1982, 269 pp.

[20] Lifshits M. A., Gaussovskie sluchainye funktsii, TViMS, Kiev, 1995, 246 pp.

[21] Martynov G. V., Kriterii omega-kvadrat, Nauka, M., 1978, 79 pp.

[22] Mischenko A. S., Fomenko A. T., Kurs differentsialnoi geometrii i topologii, Izd-vo Mosk. un-ta, M., 1980, 439 pp.

[23] Nazarov A. I., “O tochnoi konstante v obobschennom neravenstve Puankare”, Problemy matematicheskogo analiza, 2002, no. 24, 155–179

[24] Nikitin Ya. Yu., Asimptoticheskaya effektivnost neparametricheskikh kriteriev, Nauka, M., 1995, 238 pp.

[25] Piterbarg V. I., Asimptoticheskie metody v teorii gaussovskikh sluchainykh protsessov i polei, Izd-vo Mosk. un-ta, M., 1988, 175 pp.

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

[27] Pich A., Operatornye idealy, Mir, M., 1982, 536 pp.

[28] Prudnikov A. P., Brychkov Yu. A., Marichev O. I., Integraly i ryady: Elementarnye funktsii, Nauka, M., 1981, 798 pp.

[29] Fatalov V. R., “Tochnye asimptotiki bolshikh uklonenii dlya gaussovskikh mer v gilbertovom prostranstve”, Izv. NAN Armenii, 27:5 (1992), 43–57 | MR | Zbl

[30] Fatalov V. R., “Asimptotiki veroyatnostei bolshikh uklonenii gaussovskikh polei”, Izv. NAN Armenii, 27:6 (1992), 59–81 ; 28:5 (1993), 32–55

[31] Fatalov V. R., “Bolshie ukloneniya dlya gaussovskikh protsessov v gëlderovskoi norme”, Izv. RAN, 67:5 (2003), 207–224 | DOI | MR | Zbl

[32] Fatalov V. R., “Tochnye asimptotiki tipa Laplasa dlya umerennykh uklonenii raspredelenii summ nezavisimykh banakhovoznachnykh sluchainykh elementov”, Teoriya veroyatn. i ee primen., 48:4 (2003), 720–744 ; 51:3 (2006), 634–636 | DOI

[33] Fatalov V. R., “Metod Laplasa dlya malykh uklonenii gaussovskikh protsessov tipa vinerovskogo”, Matem. sb., 196:4 (2005), 135–160 | DOI | MR | Zbl

[34] Fatalov V. R., “Bolshie ukloneniya gaussovskikh mer v prostranstvakh $l^p$ i $L^p$, $p \ge 2 $”, Teoriya veroyatn. i ee primen., 41:3 (1996), 682–689 ; 51:3 (2006), 634–636 | DOI | MR | Zbl

[35] 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), 710–726 ; 51:3 (2006), 634–636 | DOI

[36] Fatalov V. R., “Tochnye asimptotiki tipa Laplasa dlya gaussovskoi mery Bogolyubova”, Teoret. i matem. fiz., 168:2 (2011), 299–340 | DOI | Zbl

[37] Fedoryuk M. V., Asimptotika: integraly i ryady, Nauka, M., 1987, 544 pp.

[38] Feinman R., Khibs A., Kvantovaya mekhanika i integraly po traektoriyam, Mir, M., 1968, 382 pp.

[39] Azencott R., “Formule de Taylor stochastique et développement asymptotique d'intégrales de Feynman”, Lecture Notes in Math., 921, 1982, 237–285 | DOI | MR | Zbl

[40] Ben Arous G., “Méthodes de Laplace et de la phase stationnaire sur l'espace de Wiener”, Stochastics, 25:3 (1988), 125–153 | DOI | MR | Zbl

[41] Bolthausen E., “Laplace approximations for sums of independent random vectors. I; II”, Probab. Theory Related Fields, 72:2 (1986), 305–318 ; 76:2 (1987), 167–206 | DOI | MR | Zbl | DOI | MR | Zbl

[42] Bolthausen E., Deuschel J.-D., Tamura Y., “Laplace approximations for large deviations of nonreversible Markov processes. The nondegenerate case”, Ann. Probab., 23:1 (1995), 236–267 | DOI | MR | Zbl

[43] Chevet S., “Gaussian measures and large deviations”, Lecture Notes in Math., 990, 1983, 30–46 | DOI | MR | Zbl

[44] Darling D. A., “On the supremum of a certain Gaussian process”, Ann. Probab., 11:3 (1983), 803–806 | DOI | MR | Zbl

[45] Davies I. M., “Laplace asymptotic expansions for Gaussian functional integrals”, Electron. J. Probab., 3:13 (1998), 1–19 | MR

[46] Donsker M. D., Varadhan S. R. S., “Asymptotic evaluation of certain Markov process expectations for large time. I; II; III, IV”, Comm. Pure Appl. Math., 28 (1975), 1–47 ; 28 (1975), 279–301 ; 29 (1976), 389–461 ; 36 (1983), 525–565 | DOI | MR | Zbl | DOI | Zbl | DOI | MR | Zbl

[47] Ellis R. S., Rosen J. S., “Asymptotic analysis of Gaussian integrals. I: Isolated minimum points”, Trans. Amer. Math. Soc., 273:2 (1982), 447–481 ; “Asymptotic analysis of Gaussian integrals. II: Manifold of minimum points”, Comm. Math. Phys., 82:2 (1981–1982), 153–181 | DOI | MR | Zbl | DOI | MR | Zbl

[48] Fatalov V. R., “The Laplace method for computing exact asymptotics of distributions of integral statistics”, Math. Methods Statist., 8:4 (1999), 510–535 | MR | Zbl

[49] Henze N., Nikitin Ya. Yu., “Watson-type goodness-of-fit tests based onthe integrated empirical process”, Math. Methods Statist., 11:2 (2002), 183–202 | MR | Zbl

[50] Kuelbs J., “Large deviation probabilities and dominating points for open convex sets: nonlogarithmic behavior”, Ann. Probab., 28:3 (2000), 1259–1279 | DOI | MR | Zbl

[51] Kusuoka S., Tamura Y., “Precise estimate for large deviation of Donsker–Varadhan type”, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 38:3 (1991), 533–565 | MR | Zbl

[52] Kusuoka S., Liang S., “Laplace approximations for sums of independent random vectors”, Probab. Theory Related Fields, 116:2 (2000), 221–238 | DOI | MR | Zbl

[53] Liang S., “Laplace approximations for sums of independent random vectors: The degenerate case”, J. Math. Sci. Univ. Tokyo, 7:2 (2000), 195–220 | MR | Zbl

[54] Lifshits M. A., “Gaussian large deviations of a smooth seminorm”, Proceedings of the Sixth USSR–Japan Symposium on Probability Theory and Mathematical Statistics, World Scientific, River Edge, 1992, 193–201 | MR | Zbl

[55] Lifshits M. A., Nazarov A. I., Nikitin Ya. Yu., “Tail behavior of anisotropic norms for Gaussian random fields”, C. R. Math. Acad. Sci. Paris, 336:1 (2003), 85–88 | DOI | MR | Zbl

[56] Pietsch A., Eigenvalues and $s$-Numbers, Cambridge Univ. Press, Cambridge, 1987, 360 pp. | MR | Zbl

[57] Podkorytova O. A., “On tail asymptotics for $ L^1$-norm of centered Brownian bridge”, Matematiche (Catania), 53:1 (1998), 3–9 | MR | Zbl

[58] Rovira C., Tindel S., “Sharp Laplace asymptotics for a parabolic SPDE”, Stochastics Stochastics Rep., 69:1–2 (2000), 11–30 | DOI | MR | Zbl

[59] Rovira C., Tindel S., “Sharp Laplace asymptotics for a hyperbolic SPDE”, Stochastic Analysis and Related Topics, VII (Kusadasi, 1998), Progr. Probab., 48, Birkhäuser, Boston, 2001, 225–244 | MR | Zbl

[60] Simon B., Functional Integration and Quantum Physics, Academic Press, New York, 1979, 296 pp. | MR

[61] Talagrand M., “Small tails for the supremum of a Gaussian processes”, Ann. Inst. H. Poincaré, 24:2 (1988), 307–315 | MR | Zbl

[62] Watson G. S., “Goodness-of-fit tests on a circle”, Biometrika, 48:1 (1961), 109–114 | MR | Zbl