Multiple maxima in multivariate samples

Abstract

Let { X n , n ⩾ 1 } be a sequence of independent random vectors in R d , d ⩾ 2 , with common continuous distribution function F. For fixed n ⩾ 2 , we say that a multiple maximum occurs among the sample points X 1 , … , X n if M n = X i for some i = 1 , … , n , with M n the componentwise sample maximum. In this paper, we investigate the asymptotic behaviour ( n → ∞ ) of the probability of observing a multiple maximum by imposing an asymptotic tail condition on F.