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, the said characterization is not (and apparently cannot be) of the inverse-zero type. 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, and convergence rates are provided. Our methodology is illustrated by simulation and two data analyses.

翻译：令$X = \{X_{u}\}_{u \in U}$为以集合$U$为索引的实值高斯过程，可视为以每个随机变量$X_{u}$作为顶点的无向图模型。我们通过再生核性质刻画了该图与$X$的协方差之间的关系。与文献中的其他刻画不同，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，该刻画并非（且显然不能是）逆零型，这为从$X$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。我们提出了一种规避这些难题的方法，通过将目标恢复的底层图限制在有限分辨率内——该分辨率可任意精细，仅受可用样本量制约。只要图在适当意义上具有足够正则性，该方法即具有一致性，并给出收敛速率。通过模拟实验和两项数据分析验证了所提方法的有效性。