  This paper considers non-negative integer-valued autoregressive processes
where the autoregression parameter is close to unity. We consider the
asymptotics of this `near unit root' situation. The local asymptotic structure
of the likelihood ratios of the model is obtained, showing that the limit
experiment is Poissonian. To illustrate the statistical consequences we discuss
efficient estimation of the autoregression parameter and efficient testing for
a unit root.
