108 BELL SYSTEM TECHNICAL JOURNAL 



The semi -invariants X,„,„ are given by the generating function 



log f(u, .) = E V^, {iurnvr + oKiu)', (iv)'] 



m,n=i mini 

 and are 



X„.„ = V ( F'"(t)G''it) dt (3.11-5) 



J— 00 



As j^ -^ 3c the distribution of / and / approaches a two dimensional normal 

 law. The approximation to this normal law may be obtained in much the 

 same manner as in section 1.6. From our assumption (3.11-2) it follows 

 that Xio and Xoi are zero. From the relation between the second moments 

 and semi-invariants X we have 



Mil = X20 + Xio = J' / F"(/) dt 



J— XI 



/+00 

 F{i)G{t) dt- (3.11-6) 



00 



/-"=" o 



M22 = X02 + Xoi = J^ / G'it) dt 



J— 00 



where the notation in the subscripts of the n's differs from that of the X's, 

 the change being made to bring it in line with sections 2.9 and 2.10 so that 

 we may write down the normal distribution at once. 



The formulas (3.11-6) are closely related to Rowland's generahzation of 

 Campbell's theorem mentioned just below equation (1.5-9). 



