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

Small deviations for two classes of Gaussian stationary processes and L p -functionals, 0 < p... 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. Problems of Information Transmission Springer Journals

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

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SP MAIK Nauka/Interperiodica
Copyright © 2010 by Pleiades Publishing, Ltd.
Engineering; Systems Theory, Control; Information Storage and Retrieval; Electrical Engineering; Communications Engineering, Networks
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