Vol. 64, No. 2, 1976

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A class of isotropic covariance functions

Yashaswini Deval Mittal

Vol. 64 (1976), No. 2, 517–538

Let {f(x),x 0} be nonnegative such that 0f(x)dx = 1. Define g(x) = f(|x|∕S|x|) 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 = {(x1xn,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)∕xn1 is nonincreasing, then the covariance function of x(t) can be computed to be r(t) = (2∕c) t∕2( 0𝜃 sinn2αdα)f(x)dx, where |t| = t, c = 0πsinn2αdα and 𝜃 = arcos(t∕2x). Let V n denote the class of covariance functions r(t) in Rn. Characterizing properties of the class V n are studied for the odd and even dimensional spaces.

Mathematical Subject Classification 2000
Primary: 60G15
Received: 30 September 1975
Published: 1 June 1976
Yashaswini Deval Mittal