Kernel density estimation for stationary random fields. (English) Zbl 1291.62083
Summary: In this paper, under natural and easily verifiable conditions, we prove the L1-convergence and the asymptotic normality of the Parzen-Rosenblatt density estimator for stationary random fields of the form \(X_k = g (\varepsilon_{k - s}, s \in \mathbb Z^d), k \in \mathbb Z^d\), where \((\varepsilon_i)_{i\in \mathbb Z^{d}}\) are independent and identically distributed real random variables and \(g\) is a measurable function defined on \(\mathbb R^{\mathbb {Z}^{d}}\). Such kind of processes provides a general framework for stationary ergodic random fields. A Berry-Esseen’s type central limit theorem is also given for the considered estimator.
MSC:
62G07 | Density estimation |
62G20 | Asymptotic properties of nonparametric inference |
60F05 | Central limit and other weak theorems |
60G60 | Random fields |