Abstract
A complex-valued stochastic process X(t) is called stationary in the wide sense (see, e.g., [32]), if its expectation is a constant, \(E\left[X(t)\right] = const,\,\,\,-\infty < t < \infty \) and the correlation function depends only on the difference \((t-s)\), that is, \(K_X (t,s)= E\left[X(t)\overline{X(s)}\right]= K_X (t-s).\) We assume that \( E \left[{|X(t)|}^2 \right] < \infty \) Let us consider a system with the memory depth \(\omega\) that maps the input stochastic process \(X(t)\) into the output stochastic process \(Y(t)\) in accordance with the following rule: \(Y(t) = \int^t_{t-\omega} X (s)g(t-s)ds, \,\,\, g(x)\epsilon L (0,\omega).\) In the optimal prediction problem one needs to find a filter \(g(t)\) such that the output process \(Y(t)\) is as close as possible to the true process \( X (t+\tau)\), where \( \tau > 0\) is a given constant. The measure of closeness is understood in the sense of minimizing the quantity \( E \left[{|X(t+\tau)-Y(t)|}^2\right] \)
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© 2012 Springer Basel
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Sakhnovich, L.A. (2012). Optimal prediction and matched filtering for generalized stationary processes. In: Levy Processes, Integral Equations, Statistical Physics: Connections and Interactions. Operator Theory: Advances and Applications, vol 225. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0356-4_4
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DOI: https://doi.org/10.1007/978-3-0348-0356-4_4
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Publisher Name: Birkhäuser, Basel
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Online ISBN: 978-3-0348-0356-4
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