A Note on Exponential Inequalities in Hilbert Spaces for Spatial Processes with Applications to the Functional Kernel Regression Model

TL;DR

This paper develops exponential inequalities for spatial processes in Hilbert spaces under dependence conditions, aiding the analysis of the functional kernel regression model's consistency.

Contribution

It introduces new exponential inequalities for spatial Hilbert space-valued processes with dependence, and applies them to establish uniform consistency in functional kernel regression.

Findings

Exponential inequalities for spatial Hilbert space processes.

Uniform consistency results for the functional kernel regression model.

Applicability under different dependence conditions.

Abstract

In this manuscript we present exponential inequalities for spatial lattice processes which take values in a separable Hilbert space and satisfy certain dependence conditions. We consider two types of dependence: spatial data under $\alpha$-mixing conditions and spatial data which satisfies a weak dependence condition introduced by Dedecker and Prieur [2005]. We demonstrate their usefulness in the functional kernel regression model of Ferraty and Vieu [2004] where we study uniform consistency properties of the estimated regression operator on increasing subsets of the underlying function space.