We derive a Berry-Esseen bound, essentially of the order of the square of the standard deviation, for the number of maxima in random samples from (0, 1)d. The bound is, although not optimal, the first of its kind for the number of maxima in dimensions higher than two. The proof uses Poisson processes and Stein's method. We also propose a new method for computing the variance and derive an asymptotic expansion. The methods of proof we propose are of some generality and applicable to other regions such as d-dimensional simplex.

http://www.stat.sinica.edu.tw/chonghi/Papers/Maxima_in_hypercubes.pdf

Maxima in hypercubes

Range-max queries on uncertain data

The concept of the Pareto set is usually encountered in the problems of the selection of a solution with respect to several criteria. In these cases instead o f an integer function one employs a vector where each Component expresses one of the criteria. The set o f the so-called conditional extremal solutions forms the Pareto set, i.e., there is no other solution which is better with respect to all criteria simultaneously. It is shown i,a [ 1 ] that in solving the discrete separable programming problems by the method of dynamic programming, the memorized variants at each step form a Pareto set. It turns out that the number o f restrictions determines the dimension of the phase space of the problem. In the present paper one finds an asymptotic estimate for the number of elements which form a Pareto set in the case o f a phase space o f an arbitrary dimension. In the same way as in [ 1 ], the components of the vector of the criteria are random variables.

A. I. Kuksa

Papers

The method of evaluating the number of conditionally optimal trajectories of discrete separable dynamic programming