  This work considers stationary vector count time series models defined via
deterministic functions of a latent stationary vector Gaussian series. The
construction is very general and ensures a pre-specified marginal distribution
for the counts in each dimension, depending on unknown parameters that can be
marginally estimated. The vector Gaussian series injects flexibility into the
model's temporal and cross-dimensional dependencies, perhaps through a
parametric model akin to a vector autoregression. We show that the latent
Gaussian model can be estimated by relating the covariances of the counts and
the latent Gaussian series. In a possibly high-dimensional setting,
concentration bounds are established for the differences between the estimated
and true latent Gaussian autocovariances, in terms of those for the observed
count series and the estimated marginal parameters. The results are applied to
the case where the latent Gaussian series is a vector autoregression, and its
parameters are estimated sparsely through a LASSO-type procedure.
