Some asymptotic properties of an estimate of the survival function under dependence conditions

Let X1, X2,..., Xn be p-dimensional random vectors coming from a strictly stationary sequence which is also subject to any one of the four standard kinds of mixing. The survival function is defined by (x) = P(X > x), where X is distributed as the X's above and the inequality X > x is to be interpreted coordinatewise. A natural estimate, n(x), for (x) is proposed and, under suitable conditions, the asymptotic normality of is established. The asymptotic behavior of the variance of n(x) and that of the covariance of (x), n(y) are also studied. Finally, it is indicated how the asymptotic normality obtained here may be used in proving asymptotic normality for a suitable version of an estimate of the hazard rate function.