A test for the independence of two Gaussian processes

A bivariate Gaussian process with mean 0 and covariance is observed in some region [Omega] of R', where {[Sigma]ij(s,t)} are given functions and p an unknown parameter. A test of H0: P = 0, locally equivalent to the likelihood ratio test, is given for the case when [Omega] consists of p points. An unbiased estimate of p is given. The case where [Omega] has positive (but finite) Lebesgue measure is treated by spreading the p points evenly over [Omega] and letting p --> [infinity]. Two distinct cases arise, depending on whether [Delta]2,p, the sum of squares of the canonical correlations associated with [Sigma](s, t, 1) on [Omega]2, remains bounded. In the case of primary interest as p --> [infinity], [Delta]2,p --> [infinity], in which case converges to p and the power of the one-sided and two-sided tests of H0 tends to 1. (For example, this case occurs when [Sigma]ij(s, t) [reverse not equivalent] [Sigma]11(s, t).)