ABSTRACT
We call a doubly indexed sequence of random variables, X m , n $ X_{m,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-math2_1.tif"/> , ( m , n ) ∈ Z 2 $ (m,n)\in \mathbb Z ^2 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math2_2.tif"/> , on ( 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-math2_3.tif"/> a random field. We say a random field X m , n $ X_{m,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-math2_4.tif"/> , ( m , n ) ∈ Z 2 $ (m,n)\in \mathbb Z ^2 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math2_5.tif"/> is a second order random field, if E | X m , n | 2 < ∞ $ E|X_{m,n}|^2<\infty $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math2_6.tif"/> , ( m , n ) ∈ Z 2 $ (m,n)\in \mathbb Z ^2 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math2_7.tif"/> . That is, X m , n ∈ L 2 ( Ω , F , P ) $ X_{m,n}\in 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-math2_8.tif"/> , ( m , n ) ∈ Z 2 $ (m,n)\in \mathbb Z ^2 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math2_9.tif"/> . A second order random field X m , n $ X_{m,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-math2_10.tif"/> , ( m , n ) ∈ Z 2 $ (m,n)\in \mathbb Z ^2 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math2_11.tif"/> is called a weakly stationary random field, if
E X m , n = C $ EX_{m,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-math2_12.tif"/> , ( m , n ) ∈ Z 2 $ (m,n)\in \mathbb Z ^2 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math2_13.tif"/> , and
cov ( X m , n , X m ′ , n ′ ) : = E ( X m , n - E X m , n ) ( X m ′ , n ′ - E X m ′ , n ′ ) ¯ $ \text{ cov}(X_{m,n},X_{m^{\prime },n^{\prime }}) := E(X_{m,n}-EX_{m,n})\overline{(X_{m^{\prime },n^{\prime }}-EX_{m^{\prime },n^{\prime }})} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math2_14.tif"/> , ( m , n ) , ( m ′ , n ′ ) ∈ Z 2 $ (m,n),(m^{\prime },n^{\prime })\in \mathbb Z ^2 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math2_15.tif"/> depends only on m - m ′ $ m-m^{\prime } $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math2_16.tif"/> and n - n ′ $ n-n^{\prime } $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/inline-math2_17.tif"/> . That is,
cov ( X m + k , n + l , X k , l ) = cov ( X m , n , X 0 , 0 ) , ( ( m , n ) , ( k , l ) ∈ Z 2 ) . $$ \begin{aligned}\text{ cov}(X_{m+k,n+l},X_{k,l}) = \text{ cov}(X_{m,n},X_{0,0}), ((m,n),(k,l)\in \mathbb Z ^2).\end{aligned} $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203709733/a3b86244-1034-43cb-9a0d-166f5a5cedfd/content/um136.tif"/>