Kernel density estimation for stationary random fields

In this paper, under natural and easily verifiable conditions, we prove the $\mathbb{L}^1$-convergence and the asymptotic normality of the Parzen-Rosenblatt density estimator for stationary random fields of the form $X_k = g\left(\varepsilon_{k-s}, s \in \Z^d \right)$, $k\in\Z^d$, where $(\varepsilon_i)_{i\in\Z^d}$ are i.i.d real random variables and $g$ is a measurable function defined on $\R^{\Z^d}$. 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.Comment: 25 page

Berry-Esseen's central limit theorem for non-causal linear processes in Hilbert space

Let $H$ be a real separable Hilbert space and $(a_k)_{k\in\mathbb{Z}}$ a sequence of bounded linear operators from $H$ to $H$. We consider the linear process $X$ defined for any $k$ in $\mathbb{Z}$ by $X_k=\sum_{j\in\mathbb{Z}}a_j(\varepsilon_{k-j})$ where $(\varepsilon_k)_{k\in\mathbb{Z}}$ is a sequence of i.i.d. centered $H$-valued random variables. We investigate the rate of convergence in the CLT for $X$ and in particular we obtain the usual Berry-Esseen's bound provided that $\sum_{j\in\mathbb{Z}}\vert j\vert\|a_j\|_{\mathcal{L}(H)}<+\infty$ and $\varepsilon_0$ belongs to $L_H^{\infty}$

On the central and local limit theorem for martingale difference sequences

Let (\Omega, \A, \mu) be a Lebesgue space and $T$ an ergodic measure preserving automorphism on $\Omega$ with positive entropy. We show that there is a bounded and strictly stationary martingale difference sequence defined on $\Omega$ with a common non-degenerate lattice distribution satisfying the central limit theorem with an arbitrarily slow rate of convergence and not satisfying the local limit theorem. A similar result is established for martingale difference sequences with densities provided the entropy is infinite. In addition, the martingale difference sequence may be chosen to be strongly mixing.Comment: Accepte pour publication dans Stochastics and Dynamic

Invariance principles for standard-normalized and self-normalized random fields

We investigate the invariance principle for set-indexed partial sums of a stationary field $(X\_{k})\_{k\in\mathbb{Z}^{d}}$ of martingale-difference or independent random variables under standard-normalization or self-normalization respectively.Comment: Submitted for publicatio

Asymptotic normality of kernel estimates in a regression model for random fields

We establish the asymptotic normality of the regression estimator in a fixed-design setting when the errors are given by a field of dependent random variables. The result applies to martingale-difference or strongly mixing random fields. On this basis, a statistical test that can be applied to image analysis is also presented.Comment: 20 page

Kernel deconvolution estimation for random fields

In this work, we establish the asymptotic normality of the deconvolution kernel density estimator in the context of strongly mixing random fields. Only minimal conditions on the bandwidth parameter are required and a simple criterion on the strong mixing coefficients is provided. Our approach is based on the Lindeberg's method rather than on Bernstein's technique and coupling arguments widely used in previous works on nonparametric estimation for spatial processes. We deal also with nonmixing random fields which can be written as a (nonlinear) functional of i.i.d. random fields by considering the physical dependence measure coefficients introduced by Wu (2005).Comment: 28 pages. arXiv admin note: text overlap with arXiv:1109.269