2. Motivation . 



Consider the set o£ nondegenerate coefficients; 





vCN) ,,(N) y(N) (N) y(N) 

 Vo+l' mo+2' mo+2' •••' niQ+r niQ+r 



where mg = ^a y and \xy denotes the largest integer less than or 

 equal to x. In the above, the constant a satisfies the inequality 

 0<a<0.5 and r is an integer constant with r^l. It will be shown that 

 under the assumptions of item 1, the set S asymptotically is multi- 

 variate normal as N ->■ °°. If the true spectrum is constant over this set 

 of r Fourier coefficients, then p(fm)/p(fm3 for the spectral lines 

 for the frequencies spanned by the band will be asymptotically distributed 



as X^/2 and the spectral density based on the average over the whole band 



2 

 will be asymptotically distributed as x^ /2r. 



The first step, then, is to prove that the set S is asymptotically 

 multivariate normal as N -*■ °° . This requires the next two listed items. 



3. A Central Limit Theorem for m-Dependent Sums (Rosen, 1967). 



Consider the double sequence of random variables: 



(1) v(i) vCD 



x^ , X2 , ..., x^ 



^(2) ^(2) (2) 



(N) (N) (N) 



A , A , . . . , A, 



That IS, the nth line of the array consists of a sequence of k,, random 



fN") ^ N 



variables. Let S be defined as: 



S^^) = f Xf ^ CC-4) 



94 



