332 BELL SYSTEM TECHNICAL JOURNAL 



Apparently there are several types of conditions which are sufficient to 

 ensure that the distribution of the resultant approaches a normal law. One 

 sufficient condition is that 



N 



-3/2 



n=l 



N 



n=l 



(2.10-3) 



The central Umit theorem tells us that the distribution of the random 

 vector (X, Y) approaches a normal law as A'^ — > co . The second moments 

 of this distribution are given by (2.1C-2). When we know the second mo- 

 ments of a normal distribution we may write down the probability density 

 function at once. Thus from section 2.9 



M = p" H, Ar' = \Mr\ ^'' "H 



Lmi2 M22J L~^i2 MnJ 



I M I = MUM22 — AI12 

 x' = [X, Y] 

 x'M~'x = I M \~\ii22X^ - 2finXY + mY') 

 The probability density is therefore 



(miiM22 — M12)~^^ [" — M22X^ — Mii^ + 2^12X7 "] , . 



27r L 2(miiM22 — )Ui2) J 



Incidentally, the second moments are related to the standard deviations 

 cTi , 0-2 of X, Y and to the correlation coefficient t of X and Y by 



Mil = 0"1 , M22 = (T2 , M12 = Tai<T2 (2.10-4) 



and the probability density takes the standard form 

 (1 - tY'" 



2ir(Tl (T2 



r 1 /X^ XY Y^\l ^ 



exp -— 2; (- - 2r — + — (2.10-5) 



L 2(1—7) \<Ti <Tia2 0'2/J 



21 This is used by Uspensky, loc. cit. Another condition analogous to the Lindeberg 

 condition is given by Cramerj^" loc. cit. 



(To be concluded) 



