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
Published: Apr 23, 2010
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