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：本文为在采样区域$R_n \subset \mathbb{R}^d$内不规则空间位置采集的空间数据，建立了一般性的序列估计量渐近理论。我们采用一种能够灵活生成不规则空间采样点的随机采样设计，该设计同时涵盖了纯增长域和混合增长域框架。具体而言，我们聚焦于一个空间趋势回归模型以及一个具有空间依赖性协变量的非参数回归模型。针对这些模型，我们研究了趋势函数与回归函数的$L^2$惩罚序列估计。作为主要结果，我们为一般序列估计量建立了一致收敛速率、$L^2$收敛速率以及多元中心极限定理。此外，我们证明了样条与小波序列估计量能够达到最优的一致收敛速率和$L^2$收敛速率，并提出了为这些估计量构建置信区间的方法。最后，我们论证了所设定的关于底层空间过程的依赖结构条件涵盖了一大类随机场，包括Lévy驱动的连续自回归与移动平均随机场。