Dependence properties of order statistics

Let X1:n⩽X2:n⩽⋯⩽Xn:n denote the order statistics of independent random variables X1,X2,…,Xn with possibly nonidentical distributions for each n. For fixed 1⩽j1y}, then Δi(y)≡P[Xj1:n>x1,Xj2:n>x2,…,Xjr:n>xr|Ai,m,y] is increasing in y, and that if Ai,m,y is either {Xi:m>y} or {Xi:m⩽y}, then Δi(y) is decreasing in i for fixed y