Let $X = \{X_{u}\}_{u \in U}$ be a real-valued Gaussian process indexed by a set $U$. It can be thought of as an undirected graphical model with every random variable $X_{u}$ serving as a vertex. We characterize this graph in terms of the covariance of $X$ through its reproducing kernel property. Unlike other characterizations in the literature, our characterization does not restrict the index set $U$ to be finite or countable, and hence can be used to model the intrinsic dependence structure of stochastic processes in continuous time/space. Consequently, this characterization is not in terms of the zero entries of an inverse covariance. This poses novel challenges for the problem of recovery of the dependence structure from a sample of independent realizations of $X$, also known as structure estimation. We propose a methodology that circumvents these issues, by targeting the recovery of the underlying graph up to a finite resolution, which can be arbitrarily fine and is limited only by the available sample size. The recovery is shown to be consistent so long as the graph is sufficiently regular in an appropriate sense. We derive corresponding convergence rates and finite sample guarantees. Our methodology is illustrated by means of a simulation study and two data analyses.

翻译：设 $X = \{X_{u}\}_{u \in U}$ 是由集合 $U$ 索引的实值高斯过程。该过程可被视为以每个随机变量 $X_{u}$ 为顶点的无向图模型。我们通过其再生核性质，以 $X$ 的协方差刻画这一图结构。与文献中其他刻画方法不同，我们的刻画不限制索引集 $U$ 为有限或可数集，因此可用于建模连续时间/空间随机过程的内在依赖结构。相应地，该刻画并不以逆协方差矩阵的零元素为表征。这给从 $X$ 的独立实现样本中恢复依赖结构（即结构估计）问题带来了新挑战。我们提出一种规避这些难题的方法论，通过恢复有限分辨率下的底层图结构——该分辨率可任意精细，仅受限于可用样本量。当图结构在适当意义下足够正则时，该恢复方法具有一致性。我们推导出相应的收敛速率与有限样本保证。通过模拟研究与两项数据分析验证了该方法的有效性。