Estimation of a Limiting Distribution Density and Its Derivatives from Observations with Weakening Dependence

We study properties of nonparametric kernel estimators for the derivatives of a multivariate distribution density. The distribution is such that the sequence of conditional distributions of dependent random variables $\varepsilon_n$ conforming with a nondecreasing $\sigma$-algebra flow $\{\mathcal F\}$ converges to this distribution. The principal part of the asymptotic mean-square error of the studied estimator with an improved rate of convergence is found. For asymptotically weakening dependence of the variables $\varepsilon_n$, the expression obtained coincides with a similar expression for the case of independent observations. The convergence with probability one and uniform asymptotic normality of the density derivative estimator under consideration is ascertained.