The authors investigate the asymptotic properties of integer-valued autoregressive model determined by the equation
where if , if , , are i.i.d. Bernoulli random variables with mean , are i.i.d. non-negative integer-valued random variables with the mean , and . Under the assumption that the second moment of the innovation is finite the authors proved that the sequence of appropriately scaled random step functions , , , formed from an unstable process () converges weakly towards a squared Bessel process determined by the stochastic differential equation , , , where , , , is a standard Wiener process. This limit process is a continuous branching process also known as square-root process or Cox-Ingersoll-Ross process. This asymptotic behavior of unstable models is quite different from that of familiar (real-valued) unstable autoregressive processes of order ( models). An application to Boston armed robberies data is presented.