Journal of Applied Mathematics and Stochastic Analysis
Volume 13 (2000), Issue 3, Pages 261-267
doi:10.1155/S104895330000023X
Negatively dependent bounded random variable probability
inequalities and the strong law of large numbers
Ferdowsi University, Faculty of Mathematical Sciences, Mashhad, Iran
Received 1 January 1998; Revised 1 January 2000
Copyright © 2000 M. Amini and A. Bozorgnia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let X1,…,Xn be negatively dependent uniformly bounded random
variables with d.f. F(x). In this paper we obtain bounds for the probabilities P(|∑i=1nXi|≥nt) and P(|ξˆpn−ξp|>ϵ) where ξˆpn is the
sample pth quantile and ξp is the pth quantile of F(x). Moreover, we
show that ξˆpn is a strongly consistent estimator of ξp under mild
restrictions on F(x) in the neighborhood of ξp. We also show that ξˆpn
converges completely to ξp.