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 X1:n⩽X2:n⩽⋯⩽Xn:n of n independent random variables X1,X2,…,Xn with possibly different distributions. For 1⩽ix1,Xj2:n>x2,…,Xjr:n>xr|Xi:n>s) is increasing in s, and that if event Ai,s is either {Xi:n>s} or {Xi:n⩽s} then P(Xj1:n>x1,Xj2:n>x2,…,Xjr:n>xr|Ai,s) is decreasing in i for fixed s. It is also shown that in this situation, if each random variable Xk has a continuous distribution function and if Ai,s is either {Xi-1:nx1,Xj2:n>x2,…,Xjr:n>xr|Ai,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].