Journal of Applied Mathematics and Stochastic Analysis
Volume 13 (2000), Issue 3, Pages 261-267

Negatively dependent bounded random variable probability inequalities and the strong law of large numbers

M. Amini and A. Bozorgnia

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.


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(|ξˆpnξp|>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.