The asymptotic distribution of the maximum of N wave crest heights for any value of the spectral width parameter

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Abstract
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 general form of the probability that any wave crest exceeds a specified crest height 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 longterm behavior of the system.

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