1032 



THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



require in addition that all states in the intervals (k8/2, (I: + 2)8/2), 



/,■ < n, are from the same "no coincidence" index set. We define q " as, 

 lli(> probability of a state 3, 6, 7, 0, or 11-15, in some interval (A"5/2, 

 (/)■ + 2)5/2), k ^ n. There are then transition probabilities from states 

 in the ?} — f to states in the /;* interval. For example. 



(n) 



and 



An) 



(n-1) 



Po 



+ (1 



= e 



-\d -m5 



(P^ 



(n-1) 







+ P4 



(n-1) 



+ V""''), 



-\5 



)(1 



„-mS 



(»-l) 



A8n 



+ (1 - e )(P; 



(«-i) 



+ Pi 



)(po 



(.-!)) + (1 



(n-1) 



)(P2 



(n-l)N 



-M«\ 



(n-1) 



+ P 



(n-1) 



in 



). 



The quantity 1 — g^"^ is then an upper bound for the probability of no 

 coincidences (upper because it is possible for a coincidence to occur in 

 the process which is not counted in this subdivision of it). The curve 

 2/5 in Fig. 8 is drawn through points at L = n8/2 computed in this 

 manner. 



To summarize the results, we see that the asymptotic formula and 

 the lower bound are both indistinguishable from the right answer; the 

 upper bounds are fairly far off. The upper bound derived by the Markov 

 process is better than that derived from the integral eciuation until 



1.0 



(n 0.6 



LU 



o 



Z 

 LU 

 Q 



U 



z 



o 

 o 



o 

 z 



U- 



o 



> 0.4 



0.6 



CD 

 < 



m 

 o 



CE 



a. 0.2 











0.1 



0.4 



0.5 



0.2 0.3 



Fig. 9 — Prol)ability of no coincidences in a 25 X 25 square; neighborhoods are 

 square. 



