In the present paper, we study the small ball probabilities in L2-norm for a family of finite-dimensional perturbations of Gaussian functions. We define three types of perturbations: noncritical, partially critical and critical; and derive small ball asymptotics for the perturbated process in terms of the small ball asymptotics for the original process. The natural examples of such perturbations appear in statistics in the study of empirical processes with estimated parameters (the so-called Durbin’s processes). We show that the Durbin’s processes are critical perturbations of the Brownian bridge. Under some additional assumptions, general results can be simplified. As an example, we find the exact L2-small ball asymptotics for critical perturbations of the Green processes (the processes whose covariance function is the Green function of the ordinary differential operator).
Similar content being viewed by others
References
M. A. Lifshits, “Asymptotic behavior of small ball probabilities,” Probab. Theory and Math. Statist. Proc. VII International Vilnius Conference (1999), pp. 453–468.
V. Li, W. and M. Shao, Q., “Gaussian processes: inequalities, small ball probabilities and applications,” Stochastic Processes: Theory and Methods, 19, 533–597 (2001).
V. R. Fatalov, “Constants in the asymptotics of small deviation probabilities for Gaussian processes and fields,” Russ. Math. Surv., 58, No. 4, 725–772 (2003).
M. A. Lifshits, Bibliography of Small Deviation Probabilities, https://airtable.com/shrMG0nNxl9SiGxII/tbl7Xj1mZW2VuYurm (2019).
G. N. Sytaya, “On some asymptotic representations of the Gaussian measure in a Hilbert space,” Theory Stoch. Proc., 2, No. 94, 93–104 (1974).
I. A. Ibragimov, “On a hitting probability of Gaussian random vector into a small ball in a Hilbert space,” Zap. Nauchn. Semin. LOMI, 85, 75–93 (1979); English transl., J. Soviet Math., 20, No. 3, 2164–2175 (1982).
V. M. Zolotarev, “Gaussian measure asymptotic in L2 on a set of centered spheres with radii tending to zero,” in: 12th Europ. Meeting of Statisticians, Varna (1979), p. 254.
J. Hoffmann-Jorgensen, L. A. Shepp, and R. M. Dudley, “On the lower tail of Gaussian seminorms,” Ann. Probab., 7, 319–342 (1979).
M. A. Lifshits, Lectures on Gaussian Processes, Springer, Berlin, Heidelberg (2012).
T. Dunker, M. A. Lifshits, and W. Linde, “Small deviation probabilities of sums of independent random variables,” Progress Probab., 43, 59–74 (1998).
W. V. Li, “Comparison results for the lower tail of Gaussian seminorms,” J. Theor. Probab., 5, No. 1, 1–31 (1992).
F. Gao, J. Hannig, and F. Torcaso, “Comparison theorems for small deviations of random series,” Electron. J. Probab., 8, No. 21, 1–17 (2003).
A. I. Nazarov and Y. Y. Nikitin, “Exact L2-small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems,” Probab. Theory Related Fields, 129, No. 4, 469–494 (2004).
A. I. Nazarov, “Exact small ball asymptotics of Gaussian processes and the spectrum of boundary value problems,” J. Theor. Probab., 22, No. 3, 640–665 (2009).
G. Birkhoff, “On the asymptotic character of the solutions of certain linear differential equations containing a parameter,” Transactions of AMS, 9, No. 2, 219–231 (1908).
G. Birkhoff, “Boundary value and expansion problems of ordinary linear differential equations,” Transactions of AMS, 9, No. 4, 373–395 (1908).
J. Tamarkin, “Sur quelques points de la theorie des equations differentielles lineaires ordinaires et sur la generalisation de la serie de Fourier,” Rendiconti del Circolo Matematico di Palermo, 34, No. 1, 345–382 (1912).
J. Tamarkin, “Some general problems of the theory of ordinary linear differential equations and expansion of an arbitrary function in series of fundamental functions,” Mathematische Zeitschrift, 27, No. 1, 1–54 (1928).
A. A. Shkalikov, “Boundary-value problems for ordinary differential equations with a parameter in the boundary conditions,” Funkt. Anal. Prilozhen., 16, No. 4, 92–93 (1982); English transl., Funct. Anal. Appl., 16, No. 4, 324–326 (1982).
A. A. Shkalikov, “Boundary-value problems for ordinary dierential equations with a parameter in the boundary conditions,” Proceed. of Petrovsky Sem., MSU Publ., 9, 190–229 (1983). English transl., J. Soviet Math., 33, 1311–1342 (1986).
P. Chigansky, M. Kleptsyna, and D. Marushkevych, “On the Karhunen–Loève expansion of Gaussian bridges,” arXiv preprint arXiv:1706.09298, (2017).
P. Chigansky and M. Kleptsyna, “Exact asymptotics in eigenproblems for fractional Brownian covariance operators,” Stochast. Processes Their Appl., 128, No. 6, 2007–2059 (2018).
P. Chigansky, M. Kleptsyna, and D. Marushkevych, “Exact spectral asymptotics of fractional processes,” arXiv preprint arXiv:1802.09045 (2018).
F. Gao and W. V. Li, “Logarithmic level comparison for small deviation probabilities,” J. Theor. Probab., 20, No. 1, 1–23 (2007).
A. I. Nazarov, “Log-level comparison principle for small ball probabilities,” Statist. Probab. Letters, 79, No. 4, 481–486 (2009).
P. Deheuvels, “A Karhunen–Loève expansion for a mean-centered Brownian bridge,” Statist. Probab. Letters, 77, No. 12, 1190–1200 (2007).
A. I. Nazarov, “On a set of transformations of Gaussian random functions,” Teor. Ver. Primen., 54, No. 2, 209–225 (2009); English transl.,Theor. Probab. Appl., 54, No. 2, 203–216 (2010).
M. Kac, J. Kiefer, and J. Wolfowitz, “On tests of normality and other tests of goodness of fit based on distance methods,” Ann. Math. Statist., 26, No. 2, 189–211 (1955).
A. I. Nazarov and Y. P. Petrova, “The small ball asymptotics in Hilbertian norm for the Kac–Kiefer–Wolfowitz processes,” Theory Probab. Appl., 60, No. 3, 482–505 (2016); English transl., Theory Probab. Appl., 60, No. 3, 460–480 (2016).
Y. P. Petrova, “Exact L2-small ball asymptotics for some Durbin processes,” Zap. Nauchn. Semin. POMI, 466, 211–233 (2017); English transl., J. Math. Sci., 244, 842–857 (2020).
M. S. Birman and M. Z. Solomyak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, 2nd ed., revised and extended. Lan’, St.Petersburg (2010); English transl. of the 1st ed.: Math. Appl. Soviet Series., vol. 5, Kluwer, Dordrecht etc., 1987.
E. C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford University Press, London (1939).
H. Bateman, “A formula for the solving function of a certain integral equation of the second kind,” Messenger Math., 37, 179–187 (1908).
S. Sukhatme, “Fredholm determinant of a positive definite kernel of a special type and its application,” Ann. Math. Statist., 1914–1926 (1972).
L. Kantorovich and V. Krylov, Approximate Methods of Higher Analysis, Interscience, New York (1960).
P. Billingsley, Convergence of Probability Measures, John Wiley & Sons (1968).
J. Durbin, “Weak convergence of the sample distribution function when parameters are estimated,” Ann. Statist., 1, No. 2, 279–290 (1973).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 501, 2021, pp. 236–258.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Petrova, Y.P. L2-Small Ball Asymptotics for a Family of Finite-Dimensional Perturbations of Gaussian Functions. J Math Sci 273, 816–831 (2023). https://doi.org/10.1007/s10958-023-06544-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-023-06544-5