Let {f(x),x ≧ 0} be
nonnegative such that ∫0∞f(x)dx = 1. Define g(x) = f(|x|∕S|x|,π) for x ∈Rn. The
n-dimensional Euclidean space is denoted by Rn, |x| is the length of the vector
x ∈ Rn and Sr,π= surface area of the n-dimensional sphere with radius r. Let W(dy)
be the (n + 1)-dimensional Gaussian white noise, i.e., for any Borel sets B and C in
Rn+1, W(B) and W(C) are mean zero Gaussian variables with variance of W(B) =
volume of B, and E(W(B)W(C)) = 0 if and only if B ∩ C = ∅. Construct the sets
At in Rn+1 as At= {(x1⋯xn,z) ∈ Rn× [0,∞)|g(x + t) > z}. Define an
n-dimensional isotropic Gaussian field as X(t) =∫AtW(dy); t ∈ Rn. X(t) has
mean zero and variance one. In addition, If it is assumed that f(x)∕xn−1 is
nonincreasing, then the covariance function of x(t) can be computed to be
r(t) = (2∕c)∫t∕2∞(∫0𝜃sinn−2αdα)f(x)dx, where |t| = t, c =∫0πsinn−2αdα and
𝜃 =arcos(t∕2x). Let Vn denote the class of covariance functions r(t) in Rn.
Characterizing properties of the class Vn are studied for the odd and even
dimensional spaces.
