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Conditions for Regularity of Stationary Random Processes

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Book cover Gaussian Random Processes

Part of the book series: Applications of Mathematics ((SMAP,volume 9))

Abstract

Let us consider a stationary narrow-sense random process ξ(t) with continuous or discrete time t. We denote, as before, by \( \mathfrak{A} \)(T) the σ-algebra of events generated by the process on the set T, that is, \( \mathfrak{A} \) (T) is the minimal σ-algebra containing events such as

$$ \left\{ {\xi ({t_1}) \in {E_1}, \ldots ,\xi ({t_s}) \in {E_s}} \right\},\,\,\,\,\,\,\,\,\,{t_1}, \ldots ,{t_s} \in T, $$
$$ P(AB) - P(A)P(B) = 0. $$
(1.1)

the E j being Borel sets on the real line.* Algebras of the form \( \mathfrak{A} \)(−∞, t) determine the past of the process (before time t), algebras of the form \( \mathfrak{A} \)(t, ∞) determine the future of the process (after time t).

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© 1978 Springer-Verlag New York Inc.

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Ibragimov, I.A., Rozanov, Y.A. (1978). Conditions for Regularity of Stationary Random Processes. In: Gaussian Random Processes. Applications of Mathematics, vol 9. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-6275-6_4

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  • DOI: https://doi.org/10.1007/978-1-4612-6275-6_4

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-6277-0

  • Online ISBN: 978-1-4612-6275-6

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