  High-dimensional time series are a core ingredient of the statistical
modeling toolkit, for which numerous estimation methods are known.But when
observations are scarce or corrupted, the learning task becomes much harder.The
question is: how much harder? In this paper, we study the properties of a
partially-observed Vector AutoRegressive process, which is a state-space model
endowed with a stochastic observation mechanism.Our goal is to estimate its
sparse transition matrix, but we only have access to a small and noisy
subsample of the state components.Interestingly, the sampling process itself is
random and can exhibit temporal correlations, a feature shared by many
realistic data acquisition scenarios.We start by describing an estimator based
on the Yule-Walker equation and the Dantzig selector, and we give an upper
bound on its non-asymptotic error.Then, we provide a matching minimax lower
bound, thus proving near-optimality of our estimator.The convergence rate we
obtain sheds light on the role of several key parameters such as the sampling
ratio, the amount of noise and the number of non-zero coefficients in the
transition matrix.These theoretical findings are commented and illustrated by
numerical experiments on simulated data.
