A Poisson approximation with applications to the number of maxima in a discrete sample

We present a Poisson approximation with applications to extreme value theory. Let X1; X2; : : : be i.i.d. and let M (1) n ¿M (2) n ¿ · · ·¿M ( j) n be the j largest order statistics. Then the asymptotic behavior of the vector (M (1) n ; : : : ; M ( j) n ) is the same as that of (M (1) N ; : : : ; M ( j) N ) where N is a random variable which is independent of X1; X2; : : : and has a Poisson distribution with mean n. The distribution of (M (1) N ; : : : ; M ( j) N ) is easy to obtain since the points X1; X2; : : : ; XN form a Poisson process on the real line. The mean measure is n dF where F is the distribution function of the Xi. We apply this to the problem of multiple maxima in discrete samples, in particular from the geometric distribution where it is known that the number of maxima has no limiting distribution. c © 1999 Elsevier Science B.V. All rights reserved