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Exact asymptotics of large deviations of stationary Ornstein-Uhlenbeck processes for L p-functionals, p > 0

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Abstract

We prove a general result on the exact asymptotics of the probability

$$P\left\{ {\int\limits_0^1 {\left| {\eta _\gamma (t)} \right|^p dt > u^p } } \right\}$$

as u → ∞, where p > 0, for a stationary Ornstein-Uhlenbeck process η γ(t), i.e., a Gaussian Markov process with zero mean and with the covariance function Eηγ(t)ηγ(s), t, s ∈ ℝ, γ > 0. We use the Laplace method for Gaussian measures in Banach spaces. Evaluation of constants is reduced to solving an extreme value problem for the rate function and studying the spectrum of a second-order differential operator of the Sturm-Liouville type. For p = 1 and p = 2, explicit formulas for the asymptotics are given.

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Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Russia

    V. R. Fatalov

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Original Russian Text © V.R. Fatalov, 2006, published in Problemy Peredachi Informatsii, 2006, Vol. 42, No. 1, pp. 52–71.

Supported in part by the Russian Foundation for Basic Research, project no. 04-01-00700.

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Fatalov, V.R. Exact asymptotics of large deviations of stationary Ornstein-Uhlenbeck processes for L p-functionals, p > 0. Probl Inf Transm 42, 46–63 (2006). https://doi.org/10.1134/S0032946006010054

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  • DOI: https://doi.org/10.1134/S0032946006010054

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