Probability pp 387-445 | Cite as

Stationary (Wide Sense) Random Sequences. L2-Theory

  • A. N. Shiryayev
Part of the Graduate Texts in Mathematics book series (GTM, volume 95)

Abstract

According to the definition given in the preceding chapter, a random sequence ξ = (ξ1, ξ2, ...) is stationary in the strict sense if, for every set B ∈ ℛ(R ) and every n ≥ 1,
$$P\left\{ {\left( {{\xi _1},{\xi _2},...} \right) \in B} \right\} = P\left\{ {\left( {{\xi _{n + 1}},{\xi _{n + 2}},...} \right) \in B} \right\}.$$
(1)
It follows, in particular, that if \(\xi _1^2 < \infty \) then Eξ n is independent of n:
$$E{\xi _n} = E{\xi _1},$$
(2)
and the covariance cov(ξ n+m ξ n ) = E(ξ n+m − Eξ n+m )(ξ n − E ξ n ) depends only on m:
$$\operatorname{cov} \left( {{\xi _{n + m}},{\zeta _n}} \right) = \operatorname{cov} \left( {{\xi _{1 + m}},{\zeta _1}} \right).$$
(3)

Keywords

Spectral Density Random Sequence Spectral Function Spectral Representation Stationary Sequence 
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 Science+Business Media New York 1984

Authors and Affiliations

  • A. N. Shiryayev
    • 1
  1. 1.Steklov Mathematical InstituteMoscowUSSR

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