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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