This paper develops a general asymptotic theory of series 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 that can flexibly generate irregularly spaced sampling sites, 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 $L^2$-penalized series estimation of the trend and regression functions. We establish uniform and $L^2$ convergence rates and multivariate central limit theorems for general series estimators as main results. 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 these 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：本文针对采样区域 $R_n \subset \mathbb{R}^d$ 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与混合递增域框架。具体而言，我们重点研究了空间趋势回归模型和含空间相依协变量的非参数回归模型。针对这些模型，我们探究了趋势函数与回归函数的 $L^2$ 惩罚序列估计方法。作为主要结果，我们建立了通用序列估计量的一致收敛率与 $L^2$ 收敛率，并推导了多元中心极限定理。此外，我们证明了样条序列估计量与小波序列估计量能达到最优的一致收敛率与 $L^2$ 收敛率，并提出了构建这些估计量置信区间的方法。最后，我们证明了对底层空间过程的相依结构条件包含一类广泛的随机场，包括莱维驱动的连续自回归滑动平均随机场。