6. A Useful Lemma. 



Let c , c , c .... be a sequence of constants and W W2 , W^, 

 be a sequence of random variables. Suppose that 



lim 



^ c 

 n -> 0° n 



c < 



and that there exists a random variable Z with finite variance such that 



the probability law for c^^W^^ converges to the probability law for Z as 



n ^ °° . Then the probability law for cW also converges to that for Z 

 as n ^ °o . 



Proof : The proof is a straightforward application of a relation given by 

 Rao (1973). In his terminology, the above lemma may be stated as: 



L L 



Cj^Wj^-> Z implies cWj^-> Z 



His relation states that this holds provided, for any e > 0, 

 lim 



c W - cW >e 

 n n n ' 







(In Rao's notation this would be stated as | Cj^Wj^ - cWj^ | -> 0). Now 



(Cn-c]2 Var(Wn) 



c W - cW > £ 

 n n n ' 







£ 



w 



> 





n ' 





1 C -c 1 

 1 n |_ 



by the Tchebichev inequality (Loeve, 1960). Since Var(Wj^) -> Var(Z) 

 and (Cj^ - c)^ ^ as n ^- "^ , it follows that: 



lim 



CnWn - cWnl > e 



n ^ 00 

 as required. Hence, the lemma is proven. 



7. Asymptotic Normality of Linear Combinations of Nondegenerate FFT 

 Coefficients . 



Let Ajj^g for s = 1,2 and mg + 1 <^ m _< mQ + r be any sequence of 

 bounded constants and define: 



,(N) 



mgH 



,(N) 



I 



m=mQ+l 



">! /N AtPn,/2 



A , nL_ 



m2 



Z 



m=mQ+l 



/N Atpn,/2 



98 



