COINCIDENCES IN POISSON PATTERNS 1021 



IV MULTIDIMENSIONAL PROBLEMS 



j 4.1 Two-Pattern Lower Bound 



} We now derive some results on the probabilities of no coincidences in 

 ! some multi-dimensional situations. The simplest one is a lower bound 

 for the case of two Poisson patterns. 



Theorem: Consider a d-dimensional region of volume T' cordaininej two 

 Poisson patterns ivith densities X and n. Let S{8) he the volume of the d-di- 

 mensional sphere of radius 5. The probability of tio coincidences between 

 the two patterns has the lower hound 



e 



Proof 



Let the pattern with density X be called the X-pattern and the other 

 the ju-pattern. Given any X-pattern of k points there will be no coinci- 

 dences provided only that a certain region T contains no points of the 

 ju-pattern. T consists of all points of the volume V which lie in any of 

 the spheres of radius 5 centered on the k points of the X-pattern. Since 

 these spheres may overlap and may extend partly outside the volume 

 T'^, we have 



volume of r ^ /.■ S{b), 



and 



Prob (no coinc, given k points) = exp ( — yu volume of 7") 



^ exp (- kn S{8)). 



Since the number, k, of points of the X-pattern has the Poisson distribu- 

 tion with mean W the (unconditional) probability of no coincidences 

 has the lower bound 



E(XF) -\v -k^siS) 

 -^-r- e e 



k 



=0 kl 



Summing the series one proves the theorem. Interchanging X and yu in 

 the theorem gives another lower V)ound. The one stated above is the 

 better of the two if X < /x. 



The difference between the lower bound and the true probability 

 comes from two sources: (a) The overlap between the k spheres; this 

 will be a small effect if \^S{28)V is small, and (b) the spheres which 

 extend partly outside the volume V; there will be relatively few such 

 spheres if only a small fraction of the volume V lies \A-ithin distance 8 



