The probability distribution of the largest value attained by a stationary random variable over a period of time, containing many oscillations, is shown to converge to a type 1 extreme value distribution for any value of the spectral width parameter. This result is derived by using recently presented methods of Galambos <1978> from the the general form of the probability that any wave crest exceeds a specified crest heights as given by Rice <1944, 1945> and further analyzed by Cartwright and Longuet-Higgins <1956>. A consequence of this generalized derivation is that several asymptotic properties of kth extremes from such distributions, previously obtained by Cartwright <1958>, using Rayleigh approximations, may be verified directly. These results illustrate the way in which the spectral width parameter affects the long-term behavior of the system.