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  • Mikhail Lifshits (a1) (a2) and Alexander Nazarov (a3)


We find logarithmic asymptotics of $L_{2}$ -small deviation probabilities for weighted stationary Gaussian processes (both for real and complex-valued) having a power-type discrete or continuous spectrum. Our results are based on the spectral theory of pseudo-differential operators developed by Birman and Solomyak.



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1. Ash, R. B. and Gardner, M. F., Topics in Stochastic Processes, Academic Press (New York, 1975).
2. Birman, M. Sh., Karadzhov, G. E. and Solomyak, M. Z., Boundedness conditions and spectrum estimates for the operators b (X)a (D) and their analogs. In Estimates and Asymptotics for Discrete Spectra of Integral and Differential Equations (Leningrad, 1989–90) (Advances in Soviet Mathematics 7 ), American Mathematical Society (Providence, RI, 1991), 85106.
3. Birman, M. Š. and Solomjak, M. Z., Asymptotics of the spectrum of pseudodifferential operators with anisotropic-homogeneous symbols. Vestnik LGU(13) 1977, 1321 (Russian). Engl. transl. Vestnik Leningrad Univ. Math. 10 (1982), 237–247.
4. Birman, M. Š. and Solomjak, M. Z., Asymptotics of the spectrum of pseudodifferential operators with anisotropic-homogeneous symbols. II. Vestnik LGU(3) 1979, 510 (Russian). Engl. transl. Vestnik Leningrad Univ. Math. 12 (1980), 155–161.
5. Birman, M. S. and Solomjak, M. Z., Spectral Theory of Self-Adjoint Operators in Hilbert Space, 2nd edn., revised and extended edn., Lan’ (St. Petersburg, 2010) (Russian). Engl. transl. of the 1st edn, Mathematics and Its Applications (Soviet Series 5), Kluwer, Dordrecht etc. 1987.
6. Bronski, J. C., Small ball constants and tight eigenvalue asymptotics for fractional Brownian motions. J. Theoret. Probab. 16(1) 2003, 87100.
7. Dunker, T., Lifshits, M. A. and Linde, W., Small deviations of sums of independent variables. In Proc. Conf. High Dimensional Probability (Progress in Probability 43 ), Birkhäuser (Basel, 1998), 5974.
8. Gao, F., Hannig, J., Lee, T.-Y. and Torcaso, F., Laplace transforms via Hadamard factorization with applications to small ball probabilities. Electronic J. Probab. 8 2003, paper 13, 1–20.
9. Gao, F., Hannig, J., Lee, T.-Y. and Torcaso, F., Exact L 2 -small balls of Gaussian processes. J. Theoret. Probab. 17(2) 2004, 503520.
10. Gengembre, S., Probabilités de petites déviations pour les processus stationnaires gaussiens. Publ. IRMA Lille 60(X) 2003, 124.
11. Gengembre, S., Petites déviations pour les processus fractionnaires. Memoire de D.E.A. Université Lille I, 2002, 19 pp.
12. Hong, S. Y., Lifshits, M. and Nazarov, A., Small deviations in L 2 -norm for Gaussian dependent sequences. Electronic Comm. Probab. 21(41) 2016, 19.
13. Kaarakka, T. and Salminen, P., On fractional Ornstein–Uhlenbeck processes. Comm. Stoch. Anal. 5(1) 2011, 121133.
14. Li, W. V. and Shao, Q.-M., Gaussian processes: inequalities, small ball probabilities and applications. In Stochastic Processes: Theory and Methods, Handbook of Statistics, Vol. 19 (eds Rao, C. R. and Shanbhag, D.), North-Holland/Elsevier (Amsterdam, 2001), 533597.
15. Lifshits, M. A., Asymptotic behavior of small ball probabilities. In Probab. Theory and Math. Statist. Proc. VII International Vilnius Conference (1998) (ed. Grigelionis, B.), VSP/TEV (Vilnius/Utrecht, 1999), 453468.
16. Lifshits, M. A., “Small deviations for stochastic processes and related topics” website. Bibliography of small deviation probabilities,
17. Lifshits, M. A. and Linde, W., Small deviations of weighted fractional processes and average non-linear approximation. Trans. Amer. Math. Soc. 357 2005, 20592079.
18. Nazarov, A. I., Log-level comparison principle for small ball probabilities. Statist. Probab. Lett. 79(4) 2009, 481486.
19. Nazarov, A. I., Exact L 2 -small ball asymptotics of Gaussian processes and the spectrum of boundary-value problems. J. Theor. Probab. 22(3) 2009, 640665.
20. Nazarov, A. I. and Nikitin, Ya. Yu., Logarithmic L 2 -small ball asymptotics for some fractional Gaussian processes. Theory Probab. Appl. 49(4) 2004, 645658.
21. Nazarov, A. I. and Pusev, R. S., Comparison theorems for the small ball probabilities of Gaussian processes in weighted L 2 -norms. Algebra & Analysis 25(3) 2013, 131146 (Russian). Engl. transl. St. Petersburg Math. J. 25(3) (2014), 455–466.
22. Neeser, F. and Massey, J., Proper complex random processes with applications to information theory. IEEE Trans. Inform. Theory 39(4) 1993, 12931302.
23. Ollila, E., On the circularity of a complex random variable. IEEE Signal Process. Lett. 15 2008, 841844.
24. Pusev, R. S., Asymptotics of small deviations of the Bogoliubov processes with respect to a quadratic norm. Theoret. Math. Phys. 165(1) 2010, 13481357.
25. Sankovich, D. P., Some properties of functional integrals with respect to the Bogoliubov measure. Theoret. Math. Phys. 126(1) 2001, 121135.
26. Zolotarev, V. M., Asymptotic behavior of Gaussian measure in 2 . J. Sov. Math. 35 1986, 23302334.
27. Zygmund, A., Trigonometrical Series, Vol. 1, Cambridge University Press (Cambridge, 1959).
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