572 CARTWKIGHT [CHAP. 15 



over the {a, 6) plane, with randomly selected e. Pierson's A'^{g, 6) is equivalent 

 to 2E{k, d){dkldG) in our notation. As in (8), the square root is inserted to show 

 the sort of stochastic convergence. 



By use of the central limit theorem (e.g. Cramer, 1946), it can be shown that 

 any of the above three models, being equivalent to the limit sum of the projec- 

 tions on a given line of a large number of vectors of given amplitude and 

 random directions, produce a Gaussian variable I,, provided certain conditions 

 hold which can usually be safely assumed. ^ This means that values of I, (x, y, t) 

 selected randomly are normally distributed, and in general groups of values 

 ^(3^1, y\, t), (,{x2, 1/2, t) . . . i,{xn, yn, t) Selected at random time intervals have a 

 joint normal distribution whose co variances are defined by the function (19). 



Some modern statisticians do not favour this approach, however. Tick (1959) 

 for example, following Doob (1953), assumes initially that a wave system (uni- 

 directional for simplicity) is a general stationary stochastic process with the 

 representation 



^(x, t,z) = { r e«(«^+P<) dU<^, iS ; z). 



where d^i{a, j8; z) is the differential of a two-dimensional random process of 

 uncorrelated increments with the orthogonal property : 



4- J rj^ / o \ Jt I ' o' M (dS{a, ^; 2) if a = a', ^ = ^' , 

 expected [d|i(a, ^ ; z) d^i{a , ^ ; 2)] = | ^ otherwise, 



and S{a, j8 ; 2) is the spectral distribution function, a generalization of our 

 E{a, k{G), 6} with constant 6. Certain properties can be deduced solely from this 

 model, but to achieve his main results, Tick makes the further assumption 

 that t,{x, t, z) is a Gaussian process, thus arriving at essentially the same system 

 as through the assumption of random phases (the random phase representation 

 is in fact implied by the Gaussian assumption). 



Rosenblatt (1957) also uses a generalized model, an integral of orthogonal 

 increments, but expressed as an infinite series of Hankel Functions in polar 

 co-ordinates. The origin of the co-ordinates is the centre of disturbance, which 

 is supposed to have the nature of a hurricane. At large distances from the 

 centre, Rosenblatt's wave system tends to the same form as Pierson's model 

 (9), though without necessarily the Gaussian (random phase) assumption. In 

 fact Rosenblatt's treatment is a rigorous mathematical exposition of the 

 physical argument roughly outlined in the beginning of this section. A still 

 more general model is developed by Kampe de Feriet (1958), who applies 

 methods of statistical mechanics to a random probability distribution of 

 "initial conditions" in a Hilbert-space. Again the Gaussian assumption is 

 invoked at a convenient stage of the argument. 



1 A sufficient set of conditions, due to Liapounoff, and described by Cramer (1946) 

 N N 



amounts to lim (2 «m^)^(2 a„2)-3 = o. 



N->-oo 1 1 



