The uniform convergence and precise asymptotics of generalized stochastic order statistics

Let X(i,n,m, k), i = 1, . . . , n, be generalized order statistics based on F. For fixed r ∈N, and a suitable counting process N(t), t > 0, we mainly discuss the precise asymptotic of the generalized stochastic order statistics X(N(n) − r + 1,N(n),m,k). It not only makes the results of Yan, Wang and Cheng [J.G. Yan, Y.B. Wang, F.Y. Cheng, Precise asymptotics for order statistics of a non-random sample and a random sample, J. Systems Sci. Math. Sci. 26 (2) (2006) 237–244] as the special case of our result, and presents many groups of weighted functions and boundary functions, but also permits a unified approach to several models of ordered random variables. © 2008 Elsevier Inc. All rights reserved