Negatively dependent bounded random variable probability inequalities and the strong law of large numbers. (2000) - Amini, M.; Bozorgnia, A.
Abstract : Let X 1, …, X n be negatively dependent uniformly bounded random variables with d.f. F ( x ) . In this paper we obtain bounds for the probabilities P ( | ∑ i = 1 n X i | ≥ n t ) and P ( | ξ ˆ p n − ξ p | > ϵ ) where ξ ˆ p n is the sample p th quantile and ξ p is the p th quantile of F ( x ) . Moreover, we show that ξ ˆ p n is a strongly consistent estimator of ξ p under mild restrictions on F ( x ) in the neighborhood of ξ p . We also show that ξ ˆ p n converges completely to ξ p .