72 BELL SYSTEM TECHNICAL JOURNAL 



In our application of this result we replace x and yhy t and I as before. 

 Then 



^ = 7 = 2 C„ COS (a'«< — ^«) 

 1 



where the primes denote differentiation with respect to /. According to the 

 central limit theorem the distribution of ^, ??, f approaches a normal law. 

 The second moments defining this law may be obtained either from the 

 above definitions of ^, 77, ^, or may be obtained from the correlation function 

 as was done in the work following equation (3.4-13). 



1^ = V'o, rf = —^0 , ^»7 = 



^ = /'(/)/''(/) = Limit I [ I'{t)r\t) dt 



T~*oa I Jo 



= Lirmt ^ [I'\T) - l"m = 



U = Limit i f mnt) dt 



T-*<x> 1 Jo 



. . .6 \1/(t) // 



= Limit , , = lAo 



Y' = Limit i [ /"(/)/"(/) ^/ 



7'-»oo i Jo 



= Limit i [ I^'\t)I{t) dt 



T-*oo 1 Jo 

 = 1^0 



where the superscript (4) represents the fourth derivative. The matrix M 

 of the moments is thus 



M = 



-^Po 

 j/'o i/'o _ 



The determinant | M \ and the cofactors of interest are 



\M\ = -^o(M^'' - yp?) (3.6-3) 



