Extremes in a random assembly

Abstract

SUMMARY The distribution of extreme values in sets of independent random variables has been very thoroughly treated (Gumbel, 1958) but few methods have been suggested for dealing with correlated variables. In this paper an approximate formula is developed for the distribution of extreme values in a normal correlated population, when the correlation matrix has dominant elements adjacent to the leading diagonal. The formula is then used to investigate extremes in a random assembly. We consider a one-dimensional interval covered by a number n of subintervals placed at random. The number, N, of subintervals covering any point is then a random variable correlated from point to point and a method is suggested whereby the extreme values of N can be estimated. Some numerical results are given, together with an asymptotic formula for large n.