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Small deviations for two classes of Gaussian stationary processes and L p-functionals, 0 < p ≤ ∞

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Abstract

Let w(t) be a standard Wiener process, w(0) = 0, and let η a (t) = w(t + a) − w(t), t ≥ 0, be increments of the Wiener process, a > 0. Let Z a (t), t ∈ [0, 2a], be a zeromean Gaussian stationary a.s. continuous process with a covariance function of the form E Z a (t)Z a (s) = 1/2[a − |t − s|], t, s ∈ [0, 2a]. For 0 < p < ∞, we prove results on sharp asymptotics as ɛ → 0 of the probabilities

$$ P\left\{ {\int\limits_0^T {\left| {\eta _a \left( t \right)} \right|^p dt \leqslant \varepsilon ^p } } \right\} for T \leqslant a, P\left\{ {\int\limits_0^T {\left| {Z_a \left( t \right)} \right|^p dt \leqslant \varepsilon ^p } } \right\} for T < 2a $$

, and compute similar asymptotics for the sup-norm. Derivation of the results is based on the method of comparing with a Wiener process. We present numerical values of the asymptotics in the case p = 1, p = 2, and for the sup-norm. We also consider application of the obtained results to one functional quantization problem of information theory.

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

    V. R. Fatalov

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Original Russian Text © V.R. Fatalov, 2010, published in Problemy Peredachi Informatsii, 2010, Vol. 46, No. 1, pp. 68–93.

Supported in part by the Russian Foundation for Basic Research, project no. 07-01-00077.

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Fatalov, V.R. Small deviations for two classes of Gaussian stationary processes and L p-functionals, 0 < p ≤ ∞. Probl Inf Transm 46, 62–85 (2010). https://doi.org/10.1134/S0032946010010060

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