COINCIDENCES IX POISSON PATTERNS 1017 



tion of points 1, • • • , k on the line is represented by a single point 

 (xi , • • ■ , Xk) in the polyhedron T in ^--dimensional space defined by the 

 inequalities 



T: ^ .Ti , • • • , ^ Xk , xi -\- X2+ ■■• + Xk -^ L, 



, and the probability distribution of the point (xi , • • • , Xk) in T is uni- 

 j form. The configurations with no coincidences lie in a smaller polyhedron 



T' consisting of all points of T for which 8 ^ xo , ■ ■ ■ , d ^ Xk . Given k, 

 I the conditional probability that there be no comcidences is the ratio of 



two A'-dimensional volumes Vol (T")/Vol (T). 



Vol (r) =0 if L S (k - 1) 8. 



For larger \-alues of L let yi — Xi , yo — X2 — 5, y^ = .Ts — 8, ■ • • , yk = 

 Xk — 8. Then T" becomes a polyhedron of the form 



T": 0^yi,0^y2,---,0^yk, yi + y2 ■ ■ ■ + yk ^ L - (A: - 1)5. 



Since the transformation from x's to y's has determinant equal to one, 

 T" has the same \^olume as T'. However, T" is now seen to be similar 

 to T but with, sides of length L — {k — 1)8 instead of L. The volume 

 ratio sought must be 



/ L- (k- 1)8 Y 



Since k has the Poisson distribution with mean XL we obtain finally 



The piecewise-analytic character of Fo(L) is evident; increasing L by 

 an amount 8 increases the upper limit on the sum bj'^ one and thereby 

 adds a new term to the analytic expression for F(L). 



24 Asymptotic Formula for F.x(L) 



Similar exact formulas could be found for all the Fx(L), but they are 

 both complicated and inconvenient for computing if L/8 becomes large. 

 It is thus natural to aim for asj^mptotic results and for bomids connected 

 with them. 



The Laplace transform of Fa(L) is given through (2-14) and (2-15) 

 above. The pole of f^■{s) with largest real part is a pole of order A^ + 1 



