This paper develops a general asymptotic theory of series ridge estimators for spatial data observed at irregularly spaced locations in a sampling region $R_n \subset \mathbb{R}^d$. We adopt a stochastic sampling design that can generate irregularly spaced sampling sites in a flexible manner including both pure increasing and mixed increasing domain frameworks. Specifically, we consider 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 and establish (i) uniform and $L^2$ convergence rates and (ii) multivariate central limit theorems for general series estimators, (iii) optimal uniform and $L^2$ convergence rates for spline and wavelet series estimators, and (iv) show that our dependence structure conditions on the underlying spatial processes cover a wide class of random fields including L\'evy-driven continuous autoregressive and moving average random fields.

翻译：本文针对不规则采样区域$R_n \subset \mathbb{R}^d$中观测的空间数据，发展了系列岭估计量的一般渐近理论。我们采用一种随机采样设计，能够灵活生成不规则分布的采样点，涵盖纯递增域与混合递增域两种框架。具体而言，我们考虑了空间趋势回归模型和含空间依赖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$L^2$惩罚系列估计，并建立了：(i)一般系列估计量的$L^2$收敛速度与一致收敛速度；(ii)多变量中心极限定理；(iii)样条与小波系列估计量的最优$L^2$收敛速度与一致收敛速度；(iv)证明所设定的空间过程依赖结构条件能覆盖包括Lévy驱动连续自回归滑动平均随机场在内的广泛随机场类。