Let {f(x),x ≧ 0} be
nonnegative such that ∫
_{0}^{∞}f(x)dx = 1. Define g(x) = f(x∕S_{x,π}) for x ∈ R_{n}. The
ndimensional Euclidean space is denoted by R_{n}, x is the length of the vector
x ∈ R_{n} and S_{r,π} = surface area of the ndimensional 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
R_{n+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
A_{t} in R_{n+1} as A_{t} = {(x_{1}⋯x_{n},z) ∈ R_{n} × [0,∞)g(x + t) > z}. Define an
ndimensional isotropic Gaussian field as X(t) = ∫
_{At}W(dy); t ∈ R_{n}. X(t) has
mean zero and variance one. In addition, If it is assumed that f(x)∕x^{n−1} is
nonincreasing, then the covariance function of x(t) can be computed to be
r(t) = (2∕c)∫
_{t∕2}^{∞}(∫
_{0}^{𝜃} sin^{n−2}αdα)f(x)dx, where t = t, c = ∫
_{0}^{π}sin^{n−2}αdα and
𝜃 = arcos(t∕2x). Let V _{n} denote the class of covariance functions r(t) in R_{n}.
Characterizing properties of the class V _{n} are studied for the odd and even
dimensional spaces.
