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

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 satisfy a weak dependence condition introduced by Dedecker and Prieur (Prob Theory Relat Fields 132(2):203–236, 2005). We demonstrate their usefulness in the functional kernel regression model of Ferraty and Vieu (Nonparametr Stat 16(1–2):111–125, 2004), where we study uniform consistency properties of the estimated regression operator on increasing subsets of the underlying function space.

Appendix

Lemma 6.1

( [27]) Let $Z_1,\ldots ,Z_n$ be real-valued non-negative random variables each a.s. bounded. Set $\alpha \,{:}{=}\,\sup _{s\in \{1,\ldots ,n\} } \alpha \left( \sigma ( Z_i: i \le k), \sigma ( Z_i: i > k) \right)$. Then $\left| {\mathbb {E}}\left[ \, \prod _{i=1}^n Z_i \, \right] - \prod _{i=1}^n {\mathbb {E}}\left[ \, Z_i \, \right] \right| \le (n-1) \, \alpha \, \prod _{i=1}^n \left\Vert Z_i \right\Vert _{\infty }$.