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)


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,
$$P\left\{ {\left( {{\xi _1},{\xi _2},...} \right) \in B} \right\} = P\left\{ {\left( {{\xi _{n + 1}},{\xi _{n + 2}},...} \right) \in B} \right\}.$$
It follows, in particular, that if \(\xi _1^2 < \infty \) then Eξ n is independent of n:
$$E{\xi _n} = E{\xi _1},$$
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:
$$\operatorname{cov} \left( {{\xi _{n + m}},{\zeta _n}} \right) = \operatorname{cov} \left( {{\xi _{1 + m}},{\zeta _1}} \right).$$


Spectral Density Random Sequence Spectral Function Spectral Representation Stationary Sequence 


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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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