LIMIT THEOREMS FOR KERNEL DENSITY ESTIMATORS IN SPACES OF ARBITRARY NATURE
Some estimators of the probability density function in spaces of arbitrary nature are used for various tasks in statistics of non-numerical data. Systematic exposition of the theory of such estimators had a start in our work [2]. This article is a direct continuation of the article [2]. We will regularly use references to conditions and theorems of the article [2], in which we introduced several types of nonparametric estimators of the probability density. We studied more linear estimators. In this article we consider particular cases - kernel density estimates in spaces of arbitrary nature. When estimating the density of the one-dimensional random variable, kernel estimators become the Parzen-Rosenblatt estimators. Asymptotic behavior of kernel density estimators in the general case of an arbitrary nature spaces are devoted to Theorem 1 - 8. Under different conditions we prove the consistency and asymptotic normality of kernel density estimators. We have studied uniform convergence. We have introduced the concept of "preferred rate differences" and studied nuclear density estimators based on it. We have also introduced and studied natural affinity measures which are used in the analysis of the asymptotic behavior of kernel density estimators. We have found the asymptotic behavior of dispersions of kernel density estimators and considered the examples including kernel density estimators in finite-dimensional spaces and in the space of square-integrable functions