We study the probability of error asymptotically for testing one Gaussian stochastic process against another when the mean vectors are zero and we have the choice between two given covariance matrices. It is shown that under certain conditions the probabilities of error form asymptotically a geometric progression with a ratio that is derived. The approach employs Laplace's method of approximating integrals and does not appeal to Fourier analysis; in this sense it can be said to be elementary.
"Large Sample Discrimination Between Two Gaussian Processes with Different Spectra." Ann. Statist. 2 (2) 347 - 352, March, 1974. https://doi.org/10.1214/aos/1176342668