Stationarity or Homogeneity of Random Fields

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Let (Ω, F, P) be a probability space on which all random objects will be defined. A filtration {Ft : 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 ∈ Rd together with a collection of distribution functions of the form Fx1,…,xn which satisfy

Fx1,…,xn(b1,…,bn) = P[Z(x1) ≤ b1,…,Z(xn) ≤ bn], b1,…,bn ∈ 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 Rd × Rd, that is if x1, . . . , xk is any collection of points in Rd, and ξ1, . . . , ξk are arbitrary real constants, then

l=1kj=1k ξlξj R(xl, xj) = ∑l=1kj=1k ξlξj E(Z(xl) Z(xj)) = E (∑j=1k ξj Z(xj))2 ≥ 0

Without loss of generality, we assumed m = 0. The property of non-negative definiteness characterizes covariance functions. Hence, given any function m : Rd → R and a non-negative definite function R : Rd × Rd → 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 Rd to the complex plane is non-negative definite if and only if it is the Fourier-Stieltjes transform of a measure F on Rd, that is the representation

R(x) = ∫Rd eix.λ dF(λ)

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

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 Zι with ι belonging to some index set I can be considered as a process in the product space (Rd, 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 ∈ Rd

• 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 ∈ Rd,

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 ∈ Rd. 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∥) = ∫Rλ=0 eitu Hd (λ ∥x∥) dG(u, λ),

where

Hd(r) = (2/r)(d – 2)/2 Γ(d/2) J(d – 2)/2 (r)

and Jm 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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