Conditions for Regularity of Stationary Random Processes

  • I. A. Ibragimov
  • Y. A. Rozanov
Part of the Applications of Mathematics book series (SMAP, volume 9)


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


Gaussian Process Orthogonal Polynomial Regularity Condition Fourier Coefficient Stationary Random Process 
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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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