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Small ball probabilities for Gaussian random fields and tensor products of compact operators


Authors: Andrei Karol', Alexander Nazarov and Yakov Nikitin
Journal: Trans. Amer. Math. Soc. 360 (2008), 1443-1474
MSC (2000): Primary 60G15; Secondary 60G60, 47A80
DOI: https://doi.org/10.1090/S0002-9947-07-04233-X
Published electronically: October 23, 2007
MathSciNet review: 2357702
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Abstract: We find the logarithmic $ L_2$-small ball asymptotics for a large class of zero mean Gaussian fields with covariances having the structure of ``tensor product''. The main condition imposed on marginal covariances is the regular behavior of their eigenvalues at infinity that is valid for a multitude of Gaussian random functions including the fractional Brownian sheet, Ornstein - Uhlenbeck sheet, etc. So we get the far-reaching generalizations of well-known results by Csáki (1982) and by Li (1992). Another class of Gaussian fields considered is the class of additive fields studied under the supremum-norm by Chen and Li (2003). Our theorems are based on new results on spectral asymptotics for the tensor products of compact self-adjoint operators in Hilbert space which are of independent interest.


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  • [Ad] R. Adler, An Introduction to Continuity, Extrema and Related Topics for General Gaussian Processes, IMS Lect. Notes - Monograph Series 12, IMS, Hayward, California, 1990. MR 1088478 (92g:60053)
  • [AnD] T.W. Anderson and D.A. Darling, Asymptotic theory of certain ``goodness-of-fit'' criteria based on stochastic processes, Ann. Math. Statist., 23 (1952), 193-212. MR 0050238 (14:298h)
  • [BNO] L. Beghin, Ya.Yu. Nikitin, E. Orsingher, Exact small ball constants for some Gaussian processes under the $ L_2$-norm, Zap. Nauchn. Sem. St.-Petersburg. Otdel. Mat. Inst. Steklov (POMI), 298 (2003), 5-21.
  • [BL] E. Belinsky and W. Linde, Small ball probabilities of fractional Brownian sheets via fractional integration operators, J. Theor. Probab., 15 (2002), 589-612. MR 1922439 (2004d:60092)
  • [BS1] M.S. Birman and M.Z. Solomyak, Quantitative analysis in Sobolev imbedding theorems and applications to spectral theory, In: Proceed. of X Summer Mathematical School, Yu.A. Mitropol'skiy and A.F. Shestopal (Eds), 1974, 5-189 (in Russian). English transl. in: AMS Transl., Ser. 2, 114, AMS, Providence, R.I., 1980. MR 562305 (80m:46026)
  • [BS2] M.S. Birman and M.Z. Solomyak, Spectral Theory of Self-adjoint Operators in Hilbert Space. Leningrad University Publishers, 1980 (in Russian). English transl. in: Math. and Its Applic. Soviet Series, 5, Dordrecht, etc.: Kluwer Academic Publishers, 1987.
  • [BS3] M.Sh. Birman and M. Solomyak, On the negative discrete spectrum of a periodic elliptic operator in a waveguide-type domain, perturbed by a decaying potential, Journ. d'Analyse Math., 83 (2001), 337-391. MR 1828497 (2002k:35226)
  • [Br] J.C. Bronski, Small ball constants and tight eigenvalue asymptotics for fractional Brownian motions, J. Theor. Probab., 16 (2003), 87-100. MR 1956822 (2004b:60105)
  • [Ca] R. Carmona, Tensor products of Gaussian measures, Lect. Notes in Math., 644 (1977), 96-124. MR 502402 (80a:60039)
  • [CC] R. Carmona and S. Chevet, Tensor Gaussian measures on $ L_p(E)$, J. Funct. Anal., 33 (1970), 297-310. MR 549116 (80j:60010)
  • [CH] C.-H. Chang and C.-W. Ha, The Green's functions of some boundary value problems via the Bernoulli and Euler polynomials, Arch. Mat., 76 (2001), 360-365. MR 1824255 (2002b:34046)
  • [CL] X. Chen and W.V. Li, Small deviation estimates for some additive processes, Proc. Conf. High Dimensional Probab. III, Progress in Probability, 55 (2003), Birkhäuser, 225-238. MR 2033891 (2004m:60115)
  • [Ch] S. Chevet, Un résultat sur les mesures gaussiennes, C.R. Acad. Sci. Paris Sér. A-B, 284 (1977), A441-A444. MR 0428413 (55:1434)
  • [Cs] E. Csáki, On small values of the square integral of a multiparameter Wiener process,. In: Statistics and Probability, Proc. of the 3rd Pannonian Symp. on Math. Stat. D.Reidel, Boston, 1982, 19-26. MR 758997
  • [DM] P. Deheuvels and G. Martynov, Karhunen - Loève expansions for weighted Wiener processes and Brownian bridges via Bessel functions, Proc. Conf. High Dimensional Probab. III, Progress in Probability, 55 (2003), Birkhäuser, 57-93. MR 2033881 (2005i:62074)
  • [De] S. Dereich, Small ball probabilities around random centers of Gaussian measures with applications to quantization, J. Theor. Probab., 16 (2003), 427-449. MR 1982037 (2004d:28033)
  • [D-MY] C. Donati-Martin and M. Yor, Fubini's theorem for double Wiener integrals and the variance of the Brownian path, Ann. Inst. H. Poincaré, Probab. Stat., 27 (1991), 181-200. MR 1118933 (92m:60072)
  • [DH-JS] R.M. Dudley, J. Hoffman-Jørgensen, L.A. Shepp, On the lower tail of Gaussian seminorms, Ann. Prob., 7 (1979), 319-342. MR 525057 (80j:60051)
  • [DLL] T. Dunker, M.A. Lifshits, W. Linde, Small deviations of sums of independent variables, In: Proc. Conf. High Dimensional Probab., Ser. Progress in Probability, Birkhäuser, 43 (1998), 59-74. MR 1652320 (2000h:60035)
  • [Fa] V.R. Fatalov, Constants in the asymptotics of small deviation probabilities for Gaussian processes and fields, Russ. Math. Surv. 58 (2003), 725-772. MR 2042263 (2005a:60052)
  • [FT] J.A. Fill and F. Torcaso, Asymptotic analysis via Mellin transforms for small deviations in $ L_2$-norm of integrated Brownian sheets, Probab. Theory and Rel. Fields, 130 (2004), 259-288. MR 2093764 (2005i:60066)
  • [GHT] F. Gao, J. Hannig, F. Torcaso, Integrated Brownian motions and exact $ L_2$-small balls, Ann. Prob., 31 (2003), 1320-1337. MR 1989435 (2004k:60104)
  • [GHLT1] F. Gao, J. Hannig, T.-Y. Lee, F. Torcaso, Laplace transforms via Hadamard factorization with applications to small ball probabilities, Electronic J. Prob., 8 (2003), 1-20. MR 1998764 (2005h:60110)
  • [GHLT2] F. Gao, J. Hannig, T.-Y. Lee, F. Torcaso, Exact $ L^2$-small balls of Gaussian processes, J. of Theor. Prob., 17 (2004), 503-520. MR 2053714 (2005d:60059)
  • [GL1] F. Gao and W.V. Li, Small ball probabilities for the Slepian Gaussian fields, Trans. Amer. Math. Soc. 359 (2007), 1339-1350. MR 2262853
  • [GL2] F. Gao and W.V. Li, Logarithmic level comparison for small deviation probabilities, J. Theoret. Probab. 20 (2007), 1-23. MR 2297848
  • [GR] I.S. Gradshteyn and I.M. Ryzhik, Tables of integrals, sums, series and products, 5th ed. Moscow, Nauka, 1971 (in Russian). English transl.: Table of integrals, series, and products. Corr. and enl. ed. by Alan Jeffrey. New York - London - Toronto: Academic Press, 1980. MR 582453 (81g:33001)
  • [GHP] S. Graf, H. Luschgy, H. Pagès, Functional quantization and small ball probabilities for Gaussian processes, J. Theor. Probab., 16 (2003), 1047-1062. MR 2033197 (2004k:60105)
  • [HN] N. Henze and Ya.Yu. Nikitin, Watson-type goodness-of-fit tests based on the integrated empirical process, Mathem. Meth. of Statist., 11 (2002), 183-202. MR 1941315 (2003i:62082)
  • [I] I.A. Ibragimov, The probability of a Gaussian vector with values in a Hilbert space hitting a ball of small radius, Zap. Nauchn. Semin. Leningrad. Otdel. Mat. Inst. Steklov (LOMI), 85 (1979), 75-93. English transl. in: J. Sov. Math. 20 (1982), 2164-2174. MR 535459 (81g:60006)
  • [KNN] A.I. Karol', A.I. Nazarov, Ya.Yu. Nikitin, Tensor products of compact operators and logarithmic $ L_2$-small ball asymptotics for Gaussian random fields, Studi Statistici No.74, Istituto di Metodi Quantitativi, Università L.Bocconi, Milano, July 2003, 30 pp.
  • [Kh] D. Khoshnevisan, Multiparameter Processes. An Introduction to Random Fields. Springer Monographs in Mathematics. NY: Springer, 2002. MR 1914748 (2004a:60003)
  • [KlB] M.L. Kleptsyna and A. Le Breton, A Cameron - Martin type formula for general Gaussian processes - a filtering approach, Stochastics and Stoch. Rep., 72 (2002), 229-250. MR 1897916 (2003g:60063)
  • [KLn] T. Kühn and W. Linde, Optimal series representation of fractional Brownian sheets, Bernoulli, 8 (2002), 669-696. MR 1935652 (2003m:60131)
  • [Li] W.V. Li, Comparison results for the lower tail of Gaussian seminorms, J. of Theor. Prob., 5 (1992), 1-31. MR 1144725 (93k:60088)
  • [LiS] W.V. Li and Q.M. Shao, Gaussian processes: inequalities, small ball probabilities and applications, In: Stochastic Processes: Theory and Methods, Handbook of Statistics, 19 (2001), C.R.Rao and D.Shanbhag (Eds), 533-597. MR 1861734
  • [Lf] M.A. Lifshits, Asymptotic behavior of small ball probabilities, In: Prob. Theory and Math. Stat., B.Grigelionis et al. (Eds), Proc. VII International Vilnius Conference, VSP/TEV, 1999, 453-468.
  • [LNN] M.A. Lifshits, A.I. Nazarov, Ya.Yu. Nikitin, Tail behavior of anisotropic norms for Gaussian random fields, C.R. Acad. Sci. Paris I, 336 (2003), 85-88. MR 1968908 (2004b:60132)
  • [LuP1] H. Luschgy and G. Pagès, Functional quantization of Gaussian processes, J. Funct. Analysis, 196 (2002), 486-531. MR 1943099 (2003i:60006)
  • [LuP2] H. Luschgy and G. Pagès, Sharp asymptotics of the functional quantization problem for Gaussian processes, Ann. Probab., 32 (2004), 1574-1599. MR 2060310 (2005d:60036)
  • [MSh] D.M. Mason and Z. Shi, Small deviations for some multi-parameter Gaussian processes, J. Theor. Probab., 14 (2001), 213-239. MR 1822902 (2001m:60091)
  • [Na1] A.I. Nazarov, On the sharp constant in the small ball asymptotics of some Gaussian processes under $ L_2$-norm, Problems of Math. Anal., 26 (2003), 179-214 (in Russian). English transl. in: J. of Math. Sci., 117 (2003), 4185-4210. MR 2027455 (2004j:60080)
  • [Na2] A.I. Nazarov, Logarithmic $ L_2$-small ball asymptotics with respect to self-similar measure for some Gaussian processes, Zap. Nauchn. Semin. St.-Petersb. Otdel. Mat. Inst. Steklov (POMI), 311 (2004), 190-213 (in Russian). To be translated in J. of Math. Sci. MR 2092208 (2005j:60078)
  • [NaNi1] A.I. Nazarov and Ya.Yu. Nikitin, Exact $ L_2$-small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems, Probab. Theory and Rel. Fields, 129 (2004), 469-494. MR 2078979 (2005d:60060)
  • [NaNi2] A.I. Nazarov and Ya.Yu. Nikitin, Logarithmic $ L_2$-small ball asymptotics for some fractional Gaussian processes, Theor. Probab. and Appl., 49 (2004), 695-711. MR 2142562 (2006b:60070)
  • [Ni] Ya. Nikitin, Asymptotic Efficiency of Nonparametric Tests, Cambridge University Press, 1995. MR 1335235 (96c:62093)
  • [PW] A. Papageorgiou and G.W. Wasilkowski, On the average complexity of multivariate problems, J. of Complexity, 6 (1990), 1-23. MR 1048027 (91b:94020)
  • [RS] M. Reed and B. Simon, Methods of Modern Mathematical Physics. V.1: Functional Analysis, New York - London: Academic Press, Inc. XVII, 1972.
  • [RWW] K. Ritter, G.W. Wasilkowski, H. Wozniakovski, Multivariate integration and approximation for random fields satisfying Sacks - Ylvisaker conditions, Ann. Appl. Probab., 5 (1995), 518-540. MR 1336881 (96d:60073)
  • [RSSh] G.V. Rosenblum, M.Z. Solomyak, M.A. Shubin, Spectral Theory of Differential Operators. Modern Problems of Mathematics, 64 (1989), Moscow, VINITI, 1-242 (in Russian). Engl. transl. in: Partial Differ. Equations VII. Encycl. Math. Sci. 64 (1994), 1-261. MR 1313735 (95j:35156)
  • [Se] E. Seneta, Regularly Varying Functions. Lect. Notes in Mathem., 508 (1976). MR 0453936 (56:12189)
  • [Sy] G.N. Sytaya, On some asymptotic representations of the Gaussian measure in a Hilbert space, In: Theory of Stochastic Processes, Kiev, 2 (1974), 93-104 (in Russian).
  • [vVW] A. van der Vaart and J.A. Wellner, Weak Convergence and Empirical Processes with Applications to Statistics, Springer Series in Statistics. NY, Springer, 1986. MR 1385671 (97g:60035)
  • [Wa] G.S. Watson, Goodness-of-fit tests on a circle, Biometrika, 48 (1961), 109-114. MR 0131930 (24:A1777)
  • [We] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen, Math. Ann., 71 (1912), 441-479. MR 1511670
  • [Z1] V.M. Zolotarev, Gaussian measure asymptotics in $ l_2$ on a set of centered spheres with radii tending to zero, In: 12th Europ. Meeting of Statisticians, Varna, 1979, 254.
  • [Z2] V.M. Zolotarev, Asymptotic behavior of Gaussian measure in $ l_2$, Problems of stability of stochastic models, Proc. Semin., Moscow, 1984, 54-58 (in Russian). English transl. in: J. Sov. Math., 24 (1986), 2330-2334. MR 859211

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Additional Information

Andrei Karol'
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetski pr., 28, St. Petersburg, 198504, Russia
Email: karol@ak1078.spb.edu

Alexander Nazarov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetski pr., 28, St. Petersburg, 198504, Russia
Email: an@AN4751.spb.edu

Yakov Nikitin
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetski pr., 28, St. Petersburg, 198504, Russia
Email: yanikit47@mail.ru

DOI: https://doi.org/10.1090/S0002-9947-07-04233-X
Keywords: Small deviations, fractional Brownian motion, Brownian sheet, Ornstein -- Uhlenbeck sheet, tensor product of operators, spectral asymptotics, slowly varying functions.
Received by editor(s): April 24, 2005
Received by editor(s) in revised form: November 22, 2005
Published electronically: October 23, 2007
Additional Notes: The authors were partially supported by RFBR Grant 04-01-00716.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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