Negative dependence in the balls and bins experiment with applications to order statistics

Dependence properties of occupancy numbers in the balls and bins experiment are studied. Applying such properties, we investigate further dependence structures of order statistics X 1 : n ⩽ X 2 : n ⩽ ⋯ ⩽ X n : n of n independent random variables X 1 , X 2 , … , X n with possibly different distributions. For 1 ⩽ i < j 1 < j 2 < ⋯ < j r ⩽ n and fixed ( x 1 , … , x r ) , we show that P ( X j 1 : n > x 1 , X j 2 : n > x 2 , … , X j r : n > x r | X i : n > s ) is increasing in s, and that if event A i , s is either { X i : n > s } or { X i : n ⩽ s } then P ( X j 1 : n > x 1 , X j 2 : n > x 2 , … , X j r : n > x r | A i , s ) is decreasing in i for fixed s. It is also shown that in this situation, if each random variable X k has a continuous distribution function and if A i , s is either { X i - 1 : n < s < X i : n } or { X i : n = s } then P ( X j 1 : n > x 1 , X j 2 : n > x 2 , … , X j r : n > x r | A i , s ) is decreasing in i for fixed s. We thus complement and extend some results in Dubhashi and Ranjan [Balls and bins: a study in negative dependence, Random Struct. Algorithms 13 (1998) 99–124] and Boland et al. [Bivariate dependence properties and order statistics, J. Multivar. Anal. 56 (1996) 75–89].