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12. Continuing Problem 11, show that Γ(1 − s) is an analytic function
of the complex variable s for |s| sufficiently small and that the con-
vergence in equation (14.9) is uniform. Consequently, the moments
of λM n − ln n converge to the moments of the extreme value density
−u
−u
−e
. Prove that this density has mean and variance
e
e
d 14. Poisson Approximation 315
ln Γ(1 − s)| s=0 = γ
ds
∞
d 2 1
ln Γ(1 − s)| s=0 = 2
2
ds k
k=1
π 2
= ,
6
where γ is Euler’s constant. (Hint: Quote whatever facts you need
about the log gamma function ln Γ(t) [12].)
14.9 References
[1] Arratia R, Goldstein L, Gordon L (1989) Two moments suffice for
Poisson approximations: the Chen-Stein method. Ann Prob 17:9–25
[2] Arratia R, Goldstein L, Gordon L (1990) Poisson approximation and
the Chen-Stein method. Stat Sci 5:403–434.
[3] Arratia R, Gordon L, Waterman MS (1986) An extreme value theory
for sequence matching. Ann Stat 14:971–993
[4] Arratia R, Waterman MS (1985) Critical phenomena in sequence
matching. Ann Prob 13:1236–1249
[5] Barbour AD, Holst L, Janson S (1992) Poisson Approximation. Oxford
University Press, Oxford
[6] D’Eustachio P, Ruddle FH (1983) Somatic cell genetics and gene fam-
ilies. Science 220:919–924
[7] Erd¨os P, R´enyi A (1970) On a new law of large numbers. J Anal Math
22:103–111
[8] Flatto L, Konheim AG (1962) The random division of an interval and
the random covering of a circle. SIAM Review 4:211–222

[9] Glaz J (1992) Extreme order statistics for a sequence of dependent
random variables. Stochastic Inequalities, Shaked M, Tong YL, editors,
IMS Lecture Notes – Monograph Series, Vol 22, Hayward, CA, pp 100–
115