Stationarity or Homogeneity of Random Fields

Let (Ω, F, P) be a probability space on which all random objects will be defined. A filtration {F_t : t ≥ 0} of σ-algebras, is fixed and defines the information available at each time t.

Random field: A real-valued random field is a family of random variables Z(x) indexed by x ∈ R^d together with a collection of distribution functions of the form F_x1,…,_xn which satisfy

F_x1,…,_xn(b₁,…,b_n) = P[Z(x₁) ≤ b₁,…,Z(x_n) ≤ b_n], b₁,…,b_n ∈ R

The mean function of Z is m(x) = E[Z(x)] whereas the covariance function and the correlation function are respectively defined as

R(x, y) = E[Z(x)Z(y)] − m(x)m(y)

c(x, y) = R(x, x)/√(R(x, x)R(y, y))

Notice that the covariance function of a random field Z is a non-negative definite function on R^d × R^d, that is if x₁, . . . , x_k is any collection of points in R^d, and ξ₁, . . . , ξ_k are arbitrary real constants, then

∑_l=1^k∑_j=1^k ξ_lξ_j R(x_l, x_j) = ∑_l=1^k∑_j=1^k ξ_lξ_j E(Z(x_l) Z(x_j)) = E (∑_j=1^k ξ_j Z(x_j))² ≥ 0

Without loss of generality, we assumed m = 0. The property of non-negative definiteness characterizes covariance functions. Hence, given any function m : R^d → R and a non-negative definite function R : R^d × R^d → R, it is always possible to construct a random field for which m and R are the mean and covariance function, respectively.

Bochner’s Theorem: A continuous function R from R^d to the complex plane is non-negative definite if and only if it is the Fourier-Stieltjes transform of a measure F on R^d, that is the representation

R(x) = ∫_R^d e^ix.λ dF(λ)

holds for x ∈ R^d. Here, x.λ denotes the scalar product ∑_k=1^d x_kλ_k and F is a bounded,  real-valued function satisfying ∫_A dF(λ) ≥ 0 ∀ measurable A ⊂ R^d

The cross covariance function is defined as R₁₂(x, y) = E[Z₁(x)Z₂(y)] − m₁(x)m₂(y)

, where m₁ and m₂ are the respective mean functions. Obviously, R₁₂(x, y) = R₂₁(y, x). A family of processes Z_ι with ι belonging to some index set I can be considered as a process in the product space (R^d, I).

A central concept in the study of random fields is that of homogeneity or stationarity. A random field is homogeneous or (second-order) stationary if E[Z(x)²] is finite ∀ x and

• m(x) ≡ m is independent of x ∈ R^d

• R(x, y) solely depends on the difference x − y

Thus we may consider R(h) = Cov(Z(x), Z(x+h)) = E[Z(x) Z(x+h)] − m², h ∈ R^d,

and denote R the covariance function of Z. In this case, the following correspondence exists between the covariance and correlation function, respectively:

c(h) = R(h)/R(o)

i.e. c(h) ∝ R(h). For this reason, the attention is confined to either c or R. Two stationary random fields Z₁, Z₂ are stationarily correlated if their cross covariance function R₁₂(x, y) depends on the difference x − y only. The two random fields are uncorrelated if R₁₂ vanishes identically.

An interesting special class of homogeneous random fields that often arise in practice is the class of isotropic fields. These are characterized by the property that the covariance function R depends only on the length ∥h∥ of the vector h:

R(h) = R(∥h∥) .

In many applications, random fields are considered as functions of “time” and “space”. In this case, the parameter set is most conveniently written as (t,x) with t ∈ R₊ and x ∈ R^d. Such processes are often homogeneous in (t, x) and isotropic in x in the sense that

E[Z(t, x)Z(t + h, x + y)] = R(h, ∥y∥) ,

where R is a function from R² into R. In such a situation, the covariance function can be written as

R(t, ∥x∥) = ∫_R ∫_λ=0^∞ e^itu H_d (λ ∥x∥) dG(u, λ),

where

H_d(r) = (2/r)^{(d – 2)/2} Γ(d/2) J_{(d – 2)/2} (r)

and J_m is the Bessel function of the first kind of order m and G is a multiple of a distribution function on the half plane {(λ,u)|λ ≥ 0,u ∈ R}.

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