960 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



Since \p is the Laplace-Stieltjes transform of the distribution of Z 

 over an interval of equiUbrium, In xp is the cumulant generating function, 

 and has the following simple form : 



rv2 



huA 



h 

 h 



f{e 



-({■+7)r 





-^T + 



- 1) _ y^T 



f +7J 



r + 7 (f + 7)' . 



(10.9) 



M is a linear function of Z, so we may obtain the cumulant generating 

 function of M in accordance with Cram^r^ (p. 187). This function is 



■r + 



r + ^ 





(10.10) 



and depends only on 6 and r. 



The mean and variance of M for an interval of equilibrium are respec- 

 tively given by 



E{M] = h, 

 var {M} = - [1 - C], with C = ^ ~ ^ , 



results which were first proved in Riordan. A normal distribution having 

 the mean and variance of M has the cumulant generating function 



-, + L' + f:(fi^- 



(10.11) 



which is to be compared to (10.10). Since var {M} goes to as T ap- 

 proaches 00 , we may expect that a suitably normalized version of Z will 

 be asymptotically normally distributed as T approaches oc . The cumu- 

 lant generating function of the normalized variable {2hhTy^ ~{Z — bT) 

 is 



Vt) 



/2 



+ 2 



exp 



■f 



1 + 



er - 



r> - 1 



m" 



+ r 



which approaches f /2 as T -^ oo . It follows that the normalized variable 

 is asymptotically normal with mean and variance 1, and that 

 (r/2by{M — b) is also asymptotically normal (0, 1). 



