1022 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



of its boundary. Hence in some cases the lower bound will be a good 

 approximation to the correct value. 



It may also be noted that no real use was made of the spherical shape 

 of the volumes S{8). If one wants to consider a point of the ^-pattern 

 to be coincident with a point of the X-pattern if it lies in some other 

 neighborhood, not of spherical shape, the same lower bound applies 

 but with S{8) replaced by the A'olume of the neighborhood. 



4.2 Single-Pattern Lower Bound 



A similar derivation in the case of a single Poisson pattern leads to: 



Theorem: Let a Poisson pattern of density X he distributed over a d-di- 



mensional region of volume V. Let S{8) be the volume of the d-dimensional 



sphere of radius 8. Then the probability of no coincidences is at least as 



large as 



g-^^{l ^ xS(8)}'"'''\ 



The theorem will follow from another bound which is slightly more 

 accurate but much more cumbersome. 



Lemma 



Ln the above theorem a lower bound is 



e-'' 1 + XF + E ^ n [1 - JS{8)/V] . (4-1) 



Proof of Lemma 



The probability sought is of the form 



Z e-''' ^' p. , (4-2) 



k Kl 



where pk is the probability that, when exactly A- points are distributed 

 at random over V, there are no coincidences. To estimate pk , imagine 

 the k points to be numbered 1,2, • • • k and placed in the region one at 

 a time. If no coincidences have been created among points 1, ••■, j 

 (which is an event of probability Pi) the probability that the addition 

 of point ./ + 1 creates no coincidence is just the probability that this 

 new point lies in none of the j spheres of radius 8 centered on points 

 1, ■••,,/. The union of these J spheres intersected with the ^■olume T^ 



