Stationary (Wide Sense) Random Sequences. L ²-Theory

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)