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THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



li-.B Upper Bounds 



Good upper bounds appear even harder to get than lower bounds. 

 One procedure i.s to divide the region V into a number of smaller cells. 

 If each cell has probability, p, of no coincidences and if there are A' 

 cells, then p'^ is the probability of no coincidence in any cell. If there is 

 no coincidence in V there will be none in any cell; hence p^ is an upper 

 bound on the probability of no coincidence in V. 



Of course, p^ is too large because of the possibility of a coincidence 

 between two points in different cells. It follows that p will be a close 

 bound only if the cell size is made large; but then p becomes hard to 

 compute. 



For example, consider self -coincidences in a single Poisson pattern in 

 a large region of area V in the plane. Cover this area with an array of 

 hexagonal cells of side 5/2 as shown in Fig. 7. The area of each hexagon 

 is 3\/3 sVS so the number of cells used will be about K = 87/3\/3 5l 

 A cell has no coincidence if it contains at most one pattern point, hence 



p = (1 + X3V3 SVS) exp - 3a/3 XS'/S. 



The upper bound is 



P = e { 1 + — g— X5 1 



which has an interesting resemblance to the lower bound 



e""'^(l + 7rX6") 



2x V7jr52 



Fig. 7 — Pattern for studying coincidences in a plane region. 



