(i.e., the sum o£ the nth row). The central limit theorem is concerned 



fNl 

 with the conditions under which the probability law of S^ -^ converges to 



the normal probability law as N -> «> . Other quantities which will be 



used in the theorem are: 



a^fs*^^^) = Variance of S*^^^ (C-5) 



Variance of x/'^^ (C-6) 



kN 



fN) 

 f (x) = probability density for X/^ '^ . (C-7) 



Theorem: If the random variables in the same row of the array are 

 m-dependent and if: 



(a) E^XjJ'^M = for all k and N, 



(b) a^fs^"^^) = 1 , N = 1,2,3,4,... , 



(c) lim Y / 

 N -> 0° Z. J 



x^ f, (x) dx = , for every e > , 



and 



k=l |x|>e 



(d) lim I o^^ <^ , 



cm 

 then the probability law for S ^ converges to a normal probability law 



having zero mean and unit variance k tends to infinity. 



Comments : Rosen gives the theorem in a more general form by stating 

 condition (c) in terms of the distribution. function and a Stieltjes inte- 

 gral. However, the above form of condition (c) in terms of the probabil- 

 ity density is sufficient for the present use. 



Proof : Given by Rosen (1967). 



4. Multivariate Central Limit Theorem (Rao, 1973). 



Let Fj^ denote the joint distribution function of the k-dimensional 



random variable ( Z^^\ Z^^\ ... , z'-'^M n = 1,2,... and F, the 

 \ n n n / An 



95 



