Negatively dependent bounded random variable probability inequalities and the strong law of large numbers (Q1841232)

Summary: Let \(X_1,\dots, X\) be negatively dependent uniformly bounded random variables with d.f. \(F(x)\). We obtain bounds for the probabilities \(P(|\sum^n_{i=1} X_i|\geq nt)\) and \(P(|\widehat\xi_{pn}- \xi_p|> \varepsilon)\) where \(\widehat\xi_{pn}\) is the sample \(p\)th quantile and \(\xi_p\) is the \(p\)th quantile of \(F(x)\). Moreover, we show that \(\widehat\xi_{pn}\) is a strongly consistent estimator of \(\xi_p\) under mild restrictions on \(F(x)\) in the neighborhood of \(\xi_p\). We also show that \(\widehat\xi_{pn}\) converges completely to \(\xi_p\).