Multiple maxima in multivariate samples

Let {Xn,n⩾1} be a sequence of independent random vectors in Rd,d⩾2, with common continuous distribution function F. For fixed n⩾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→∞) of the probability of observing a multiple maximum by imposing an asymptotic tail condition on F.