distribution function of the linear function X, Z + X_Z + ... 



1 n 2 n 



+ X.Z . Also let F be the joint distribution function of a 



k-dimensional random variable Z , Z , . . . , Z . If for each vector 



X, F, ^ F,, the distribution function of A^Z^-*^-^ + X-Z*- •' + ...+ X Z*^ ^ 

 Xn X T T I- 



then Fj^ -^ F. 



Proof : See Rosen (1967). 



5. Some Conditions for which (c) in the Theorem of Item 3 Holds . 



Suppose that c , are constants uniformly bounded in n and k, 



X , = c , Y , /v^ , 

 nk nk nk 



and the probability density of Y , is denoted by g i^ (x) • If: 



(a) there exists positive constants a, b, and c such that 



g^k*^^^ 1 ae~ 1^1 , if Iy|>c 



uniformly in n and k, or 



(b) there exists a positive constant A such that 



?nk(y) = , if |y|>A 



uniformly in n and k. 

 Then for any e>0, 



11 



,2 



L = lim \ / x^ f , (x) dx = . 



k=l |x|>e 



Proof: The probability density for Y , expressed in terms of f , (x) is: 



Snk^>') =(lSkl/ ^jfnklw/'^) • 



96 



