A test for the independence of two Gaussian processes

A bivariate Gaussian process with mean 0 and covariance

$\text{Σ(s, t, p)=}\text{Σ}{}_{11}\text{(s, t)}\text{ρΣ}{}_{12}\text{(s, t)}\text{ρΣ}{}_{21}\text{(s, t)}\text{Σ}{}_{22}\text{(s, t)}$

is observed in some region Ω of R′, where {Σ_ij(s,t)} are given functions and p an unknown parameter. A test of H₀: 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 $\text{Ω}{}^2$, remains bounded. In the case of primary interest as p → ∞, Δ_2,p → ∞, in which case $\text{p}\text{?}$ 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 Σ_ij(s, t) ≡ Σ₁₁(s, t).)