Finally condition (c) will be satisfied if the probability density 



for the water level elevations obey the bounding conditions (a) and (b) 



in item 5. This was assumed in item 1. (Note: B = lim B,, + 1 will 



n -> oo N 



provide the uniform bound for c , needed in item 5.) 



Thus, all conditions are satisfied and T /a(T ^) converges in 

 law to a zero mean, unit-variance normal probability law. Since 



mn+r 



"0 



1 



m=mn+l 



"'(V- Z (*^l**m2) = 



"0 



fN) 

 as N -^ °° , it follows from item 6 that the probability law of T /a 



converges to zero mean, unit-variance normal probability law. This 



completes the proof. 



8. Asymptotic Normality of the FFT Coefficients . 



Let ( Ujjj ' ^m ) ^°^ *" " "'O'^-'-' '"o"^^' ■"■' "^"^^ ^® ^ ^^^ °^ 

 nondegenerate FFT coefficients. Then the multivariate probability law 

 for 



U(N) ^(N) 



■v/Np At/2 s/up At/2 



, mQ+1 ^ m <^ '"O"*'^ 



will be a multivariate normal with covariance matrix equal to an identity 

 matrix and mean vector having all components equal to zero. 



Proof: In items 4 and 7. 



9. Asymptotic Chi-Squared Distribution for Spectral Estimates . 



For the range of m-values, nig+1 £ m <^ '"o'^-'^' suppose that p = P 

 is a nonzero constant. Let: 



m^+r 



"0 



} 



m=mn+l 



?■? Z ("^'vO/N^t 



'0^ 



101 



