Series ridge regression for spatial data on $\mathbb{R}^d$

This paper develops a general asymptotic theory of series ridge estimators for spatial data collected at irregularly spaced locations within a sampling region $R_n \subset \mathbb{R}^d$. We employ a stochastic sampling design capable of generating irregularly spaced sampling sites flexibly, encompassing both pure increasing and mixed increasing domain frameworks. Specifically, we focus on a spatial trend regression model and a nonparametric regression model with spatially dependent covariates. For these models, we investigate the $L^2$-penalized series estimation of the trend and regression functions. As main results, we establish uniform and $L^2$ convergence rates and multivariate central limit theorems for general series estimators. Additionally, we show that spline and wavelet series estimators achieve optimal uniform and $L^2$ convergence rates, and propose methods for constructing confidence intervals for spline and wavelet estimators. Finally, we demonstrate that our dependence structure conditions on the underlying spatial processes include a broad class of random fields, including L\'evy-driven continuous autoregressive and moving average random fields.