From this, leakage would always be to the immediate neighboring lines. 

 The outlier lines in Table 3 are always separated by one or more small 

 lines in each case. Hence, one has to conclude that the cases with two 

 or more outlier spectral lines cannot be explained by a single wave train 

 with FFT leakage. 



XIII. WHY DOES CHI-SQUARED WORK FOR HURRICANE WAVES? 



One of the mysteries arising from the data is the surprisingly good 

 probabilities arising from the chi-squared derivations. If linear wave 

 theory was holding and the seas were Gaussian, this would be expected 

 (Borgman, 1972, 1973). However, the hurricane waves were decidedly non- 

 linear. The waves in many of the records have been plotted by computer 

 and examined visually. The nonlinearities are really there. Why does the 

 the chi-squared work so well? 



Investigation of this question led to a central limit theorem for 

 dependent random variables which showed that the chi-squared relations 

 hold exactly as N tends to infinity even for a non-Gaussian sea surface. 

 The primary limitation is that water level elevations at the recorder 

 separated by more than a certain constant time interval should be statis- 

 tically independent of each other. The amount of the separation required 

 to achieve independence is unimportant in the validity of the theorem 

 although it will affect the speed of convergence to the asympototic 

 result. Time sequences with above dependency properties are said to be 

 "m-dependent ," in statistical terminology. 



A sufficient condition, then, for the central limit theorem to hold 

 is that the probability density of the water level elevation measured 

 from mean water level satisfies at least one of two "tail" conditions. 

 The density, f(n), should be such that there exists positive constants 

 a, b, and c, and a positive integer n, such that: 



f(n3 1 a|y|" e'^l^l , |y| > c , (69) 



or alternately, there exists a positive constant A such that: 



f(ri) =0 , if |y| > A . (70) 



The second condition is a special case of the first since if the second 

 condition holds then c = A and any a, b, and n values will permit the 

 first condition to be satisfied. 



Equation (69) is not an unreasonable restriction. For low seas, the 

 sea surface has been found to be normally distributed. As the wave heights 

 increase, a gamma density might be a reasonable guess as to the proper 

 probability law. Both of these densities satisfy equation (69). In fact, 

 from a practical viewpoint, no one seriously suggests that water level 

 elevations can be infinite as required by the normal or the gamma densities, 

 In fact, there will be a large value of A (e.g., A = water depth) such 

 that equation (70) will hold. 



81 



