COINCIDENCES IN POISSON PATTERNS 1011 



for some h > a (note that the left hand side of the preceding inequahty 

 does not approach as ?/ approaches ± co ) . 

 The pole of /(s) at s = —a contributes to F{L) a dominant term 



F(T) ~ X' + m' - (X + M)a + 2X^6"^'^'''°^^ -aL ^ jQS 



^ ^ ^ (\-\- n - a)[K + M - 2a + 25(X - a)(n -a)] 



In (1-10) the error is 0(exp — bL) for large L. 

 When 8 is small, we find a = 2\n8 + 0(5") and (1-10) becomes 



F(L) ^ [1 + 0(6')] exp - [2Xm5 + 0(5')]L. (1-11) 



It is interesting to note that a simple heuristic argument also leads to a 

 formula like (1-11). When 5 is small and L is large, one expects that the 

 intervals of length 25 which contain points of Pattern No. 1 at their cen- 

 ters will comprise a total length near (XL) (25) of the line segment (0, L) . 

 The probabiUty that a set of length 2XL5 shall be free of points of Pat- 

 tern No. 2 is exp — 2Xm5L. 



1.3 Bounds 



In this section we derive some relatively simple expressions which are 

 good upper and lower bounds on F(L) . Both bounds have the same func- 

 tional form: 



K{A, B; L) = h±±J^ e-- + fi _ ^^^ ^ ^^ ) e'''^^''. (1-12) 

 X-f-ju — a \ \ -\- ti — a/ 



In (1-12), a is again the smallest real solution of D( — a) = 0. .4 and B 

 are positive constants which are related by 



A _ M -(X+M-a)S _ X — g (X+^_a)5 (1-13) 



B n — a X 



K{A, B; L) becomes an upper bound or a lower bound depending on ad- 

 ditional restrictions which will be placed on A and B. 



To get the lower bound, we restrict .4 and B by the inequalities 



A < e'"-"', B < e'"-^'', (1-14) 



and 



A<(l-f\e''\ B<(\-^^e'\ (1-15) 



We first prove that (1-13), (1-14), and (1-15) imply 



Pi(L) > Ae-''\ PoXL) > Be'"''. (1-16) 



