The spectral density function describes the second-order properties of a stationary stochastic process on $\mathbb{R}^d$. This paper considers the nonparametric estimation of the spectral density of a continuous-time stochastic process taking values in a separable Hilbert space. Our estimator is based on kernel smoothing and can be applied to a wide variety of spatial sampling schemes including those in which data are observed at irregular spatial locations. Thus, it finds immediate applications in Spatial Statistics, where irregularly sampled data naturally arise. The rates for the bias and variance of the estimator are obtained under general conditions in a mixed-domain asymptotic setting. When the data are observed on a regular grid, the optimal rate of the estimator matches the minimax rate for the class of covariance functions that decay according to a power law. The asymptotic normality of the spectral density estimator is also established under general conditions for Gaussian Hilbert-space valued processes. Finally, with a view towards practical applications the asymptotic results are specialized to the case of discretely-sampled functional data in a reproducing kernel Hilbert space.

翻译：谱密度函数描述了$\mathbb{R}^d$上平稳随机过程的二阶性质。本文考虑取值于可分离Hilbert空间的连续时间随机过程的谱密度的非参数估计。我们的估计量基于核平滑方法，可应用于多种空间采样方案，包括数据在不规则空间位置观测的情形。因此，该方法在空间统计学中具有直接应用价值，因为不规则采样数据在该领域自然产生。在混合域渐近框架下的一般条件下，我们获得了估计量的偏差和方差的速率。当数据在规则网格上观测时，估计量的最优速率与服从幂律衰减的协方差函数类的极小极大速率相匹配。同时，对于高斯Hilbert空间值过程，在一般条件下建立了谱密度估计量的渐近正态性。最后，面向实际应用，我们将渐近结果特化到再生核Hilbert空间中离散采样函数数据的情形。