Let X_n, 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 ϕ_k(s, t)=P(X₁ s, X_k+1 t)-F(s)F(t). We prove the almost sure consistency for the histogram to estimate each ϕ_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^-1/3log^2/3n for the estimator of ϕ_k and of order n^-1/3log^5/3n for the estimator of the covariance function.