Multiple maxima in multivariate samples

Let {Xn,n[greater-or-equal, slanted]1} be a sequence of independent random vectors in , with common continuous distribution function F. For fixed n[greater-or-equal, slanted]2, we say that a multiple maximum occurs among the sample points X1,...,Xn if Mn=Xi for some i=1,...,n, with Mn the componentwise sample maximum. In this paper, we investigate the asymptotic behaviour (n-->[infinity]) of the probability of observing a multiple maximum by imposing an asymptotic tail condition on F.