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.

翻译：本文为采样区域 $R_n \subset \mathbb{R}^d$ 内不规则分布的空间数据建立了序列岭估计的一般渐近理论。我们采用一种能够灵活生成不规则采样点的随机抽样设计，该设计同时涵盖纯递增域和混合递增域框架。具体而言，我们关注空间趋势回归模型和具有空间相依协变量的非参数回归模型。针对这些模型，我们研究趋势函数与回归函数的 $L^2$ 惩罚序列估计。作为主要结果，我们建立了通用序列估计的一致收敛率与 $L^2$ 收敛率，以及多元中心极限定理。此外，我们证明了样条估计与小波估计可达到最优的一致收敛率与 $L^2$ 收敛率，并提出构建样条估计与小波估计置信区间的方法。最后，我们证明所提出的空间过程相依结构条件包含广泛的随机场类别，包括Lévy驱动的连续自回归移动平均随机场。