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

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.