Weakly Stationary Sequences
Vidyadhar S. Mandrekar, David A. Redett · 2017
Let Ω $ \Omega $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math1_1.tif"/> be a nonempty set, F $ \mathcal F $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math1_2.tif"/> a σ $ \sigma $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math1_3.tif"/> -algebra of subsets of Ω $ \Omega $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math1_4.tif"/> , and P a probability measure defined on F $ \mathcal F $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math1_5.tif"/> . We will call the triple ( Ω , F , P ) $ (\Omega ,\mathcal F ,P) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math1_6.tif"/> a probability space. All complex-valued, F $ \mathcal F $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math1_7.tif"/> -measurable functions X on Ω $ \Omega $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math1_8.tif"/> will be called random variables. For a random variable X , we will write EX to denote the integral ∫ Ω X d P $ \displaystyle \int _\Omega X\,dP $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math1_9.tif"/> . A sequence, X n $ X_n $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math1_10.tif"/> , n ∈ Z $ n\in \mathbb Z $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math1_11.tif"/> , of random variable will be called a random sequence. We say a random sequence X n $ X_n $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math1_12.tif"/> , n ∈ Z $ n\in \mathbb Z $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math1_13.tif"/> is a second order sequence, if E | X n | 2 , n ∈ Z $ n\in \mathbb Z $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math1_15.tif"/> . Recall that such a sequence is contained in L 2 ( Ω , F , P ) $ L^2(\Omega ,\mathcal F ,P) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math1_16.tif"/> . A second order sequence X n $ X_n $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math1_17.tif"/> , n ∈ Z $ n\in \mathbb Z $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math1_18.tif"/> is called a weakly stationary sequence, if E X n = C $ EX_n=C $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math1_19.tif"/> , n ∈ Z $ n\in \mathbb Z $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math1_20.tif"/> , and cov ( X n , X m ) : = E ( X n - E X n ) ( X m - E X m ) ¯ $ \text{ cov}(X_n,X_m) := E(X_n-EX_n)\overline{(X_m-EX_m)} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math1_21.tif"/> , n , m ∈ Z $ n,m\in \mathbb Z $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math1_22.tif"/> depends only on n - m $ n-m $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math1_23.tif"/> . That is, 1.1 cov ( X n + k , X k ) = cov ( X n , X 0 ) , n , k ∈ Z . $$ \begin{aligned} \text{ cov}(X_{n+k},X_k) = \text{ cov}(X_n,X_0), n,k\in \mathbb Z .\end{aligned} $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/math1_1.tif"/>