ON THE VARIANCE OF THE NUMBER OF MAXIMA IN RANDOM VECTORS AND ITS APPLICATIONS

Let X = {x1,x2, . . . ,xn} be a set of independent and identically distributed (iid) random vectors in Rd. A point xi = (xi1, . . . , xid) is said to be dominated by xj if xik xjk for all k and xi` > xj` for some `. Then the optimal variants constitute the so-called Pareto set of X, that is, the set of all xi which are not “≺” by others. The Pareto set has been actively investigated since the seventies, notably in Russia; see the survey paper by Sholomov (1983). Under the assumptions that x1, . . . ,xn are iid and that the components of each vector are identically and continuously distributed, the Pareto set is identical to the set of maxima. In the sequel, all (with only one exception) results concerning the random variables Kn,d mentioned in this paper are under the above assumptions. Dominance is clearly one of the natural order relations in multivariate observations. Thus, the