  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.
