Abstract. An inhomogeneous first-order integer-valued autoregressive (INAR(1)) process is investigated, where the autoregressive type coefficient converges to one and the immigration mean tends to zero at appropriate speeds. It is shown that the process converges weakly to a Poisson or a compound Poisson distribution.

AMS Subject Classification (1991): 60J80; 60J27, 60J85

Keyword(s): Integer--valued time series, INAR(1) model, nearly unstable model, Poisson approximation, Galton--Watson process

Received March 22, 2007, and in revised form June 29, 2007. (Registered under 6462/2009.)