Limit theorems for kernel density estimators under dependent samples

TL;DR

This paper establishes limit theorems and convergence rates for kernel density estimators using dependent samples, extending classical results for independent data to dependent, mixing sequences.

Contribution

It introduces a moment inequality for mixing dependent variables and proves that optimal convergence rates for i.i.d. data also hold under dependence.

Findings

Central limit theorems for kernel density estimators and their distribution functions.

Convergence rates in sup-norm and L^p norms for dependent samples.

Almost sure convergence rates over compact sets and the entire real line.

Abstract

In this paper, we construct a moment inequality for mixing dependent random variables, it is of independent interest. As applications, the consistency of the kernel density estimation is investigated. Several limit theorems are established: First, the central limit theorems for the kernel density estimator $f_{n,K}(x)$ and its distribution function are constructed. Also, the convergence rates of $\|f_{n,K}(x)-Ef_{n,K}(x)\|_{p}$ in sup-norm loss and integral $L^{p}$-norm loss are proved. Moreover, the a.s. convergence rates of the supremum of $|f_{n,K}(x)-Ef_{n,K}(x)|$ over a compact set and the whole real line are obtained. It is showed, under suitable conditions on the mixing rates, the kernel function and the bandwidths, that the optimal rates for i.i.d. random variables are also optimal for dependent ones.