Asymptotic behaviour of estimators of the parameters of nearly unstable INAR(1) models

A sequence of first-order integer-valued autoregressive type (INAR(1)) processes is investigated, where the autoregressive type coefficients converge to 1. It is shown that the limiting distribution of the joint conditional least squares estimators for this coefficient and for the mean of the innovation is normal. Consequences for sequences of Galton-Watson branching processes with unobservable immigration, where the mean of the offspring distribution converges to 1 (which is the critical value), are discussed.