302 BELL SYSTEM TECHNICAL JOURNAL 



1.5 Extension of Campbell's Theorem 



In section 1.2 we have stated Campbell's theorem. Here we shall give 

 an extension of it. In place of the expression (1.2-1) for the I{t) of the shot 

 effect we shall deal with the current 



+ 00 



I 



m = E a,F{t - h) (1.5-1) 



where F{t) is the same sort of function as before and where • • • ai , a^ , • • • 

 ak , • • • are independent random variables all having the same distribution. 

 It is assumed that all of the moments a"^ exist, and that the events occur at 

 random 



The extension states that the nth semi-invariant of the probabiUty density 

 P{I) of /, where / is given by (1.5-1), is 



\n= v^- I [F{t)Tdt (1.5-2) 



J— 00 



where v is the expected number of events per second. The semi-invariants 

 of a distribution are defined as the coefficients in the expansion 



log. (ave. e'^") = E -! {iuf + o(w^) (1.5-3) 



n=i n\ 



i.e. as the coefficients in the expansion of the logarithm of the characteristic 

 function. The X's are related to the moments of the distribution. Thus if 

 Wi , W2 , • • • denote the first, second • • • moments about zero we have 



N 



ave. e = 1 + Z^ — i (*w) + o{u ) 



By combining this relation with the one defining the X's it may be shown that 

 / = wi = Xi 



/2 = W2 = X2 + XiWi 



P = mz = X3 + 2X2W1 + X1W2 



It follows that Xi = / and X2 = ave. {I — I) . Hence (1.5-2) yields the 

 original statement of Campbell's theorem when we set n equal to one and 

 two and also take all the a's to be unity. 



The extension follows almost at once from the generalization of expression 

 (1.4-7) for the probability density P{I). By proceeding as in section 1.4 

 and identifying Xk with akF{t — tk) we see that 



ave, e"*" =7^,1 q{a) da I exp [iuaF{t — tk)] dtk 

 1 J-00 Jo 



