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Conditions for Regularity of Stationary Random Processes

  • I. A. Ibragimov
  • Y. A. Rozanov
Chapter
Part of the Applications of Mathematics book series (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).

Keywords

Gaussian Process Orthogonal Polynomial Regularity Condition Fourier Coefficient Stationary Random Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1978

Authors and Affiliations

  • I. A. Ibragimov
    • 1
  • Y. A. Rozanov
    • 2
  1. 1.LomiLeningradUSSR
  2. 2.V.A. Steklov Mathematics InstituteMoscowUSSR

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