  This paper develops the theory and methods for modeling a stationary count
time series via Gaussian transformations. The techniques use a latent Gaussian
process and a distributional transformation to construct stationary series with
very flexible correlation features that can have any pre-specified marginal
distribution, including the classical Poisson, generalized Poisson, negative
binomial, and binomial structures. Gaussian pseudo-likelihood and implied
Yule-Walker estimation paradigms, based on the autocovariance function of the
count series, are developed via a new Hermite expansion. Particle filtering and
sequential Monte Carlo methods are used to conduct likelihood estimation.
Connections to state space models are made. Our estimation approaches are
evaluated in a simulation study and the methods are used to analyze a count
series of weekly retail sales.
