A test for the independence of two Gaussian processes

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A bivariate Gaussian process with mean 0 and covariance Σ(s, t, p)= Σ 11 (s, t) ρΣ 12 (s, t) ρΣ 21 (s, t) Σ 22 (s, t) is observed in some region Ω of R ′, where { Σ ij ( s , t )} are given functions and p an unknown parameter. A test of H 0 : p = 0, locally equivalent to the likelihood ratio test, is given for the case when Ω consists of p points. An unbiased estimate of p is given. The case where Ω has positive (but finite) Lebesgue measure is treated by spreading the p points evenly over Ω and letting p → ∞. Two distinct cases arise, depending on whether Δ 2, p , the sum of squares of the canonical correlations associated with Σ ( s , t , 1) on Ω 2 , remains bounded. In the case of primary interest as p → ∞, Δ 2, p → ∞, in which case p ̂ converges to p and the power of the one-sided and two-sided tests of H 0 tends to 1. (For example, this case occurs when Σ ij ( s , t ) ≡ Σ 11 ( s , t ).)