Convergence rates for the estimation of two-dimensional distribution functions under association and estimation of the covariance of the limit empirical process

Abstract

Let Xn, n=1, be an associated and strictly stationary sequence of random variables, having marginal distribution function F. The limit in distribution of the empirical process, when it exists, is a centred Gaussian process with covariance function depending on terms of the form ?k(s, t)=P(X1 s, Xk+1 t)-F(s)F(t). We prove the almost sure consistency for the histogram to estimate each ?k and also to estimate the covariance function of the limit empirical process, identifying, for both, uniform almost sure convergence rates. The convergence rates depend on a suitable version of an exponential inequality. The rates obtained, assuming the covariances to decrease geometrically, are of order n-1/3log2/3nfor the estimator of ?k and of order n-1/3log5/3nfor the estimator of the covariance function.