A test for the independence of two Gaussian processes

AbstractA bivariate Gaussian process with mean 0 and covariance Σ(s, t, p)=Σ11(s, t)ρΣ12(s, t)ρΣ21(s, t)Σ22(s, t) is observed in some region Ω of R′, where {Σij(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 Δ2,p, the sum of squares of the canonical correlations associated with Σ(s, t, 1) on Ω2, remains bounded. In the case of primary interest as p → ∞, Δ2,p → ∞, in which case p̂ 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 Σij(s, t) ≡ Σ11(s, t).

Expressions for the normal distribution and repeated normal integrals

We give a new expression for Mills ratio and five new expressions for repeated integrals of the univariate normal density, or equivalently for the Hermite functions. The Hermite functions are shown to be the negative moments of x+iN where N is a unit normal random variable and . This extends an earlier result that the Hermite polynomials are the positive moments of x+iN. We also give the derivatives of Mills' ratio and its inverse.Normal distribution Repeated integrals Mills' ratio Hermite functions Airey functions Laplace's expansion Univariate