The INAR(1) processes with coefficients alpha(n) = 1 - c/kn , where c > 0 is a fixed constant and {k(n)}(n is an element of N) is a deterministic sequence growing to infinity at a slower rate than n, which are often referred to as nearly unstable INAR processes with moderate deviations from a unit root. We consider some basic properties of the processes and obtain the conditional least squares estimation of the coefficient alpha(n) , which converges to a normal distribution at speed n(1/2)k(n). The simulation study provides numerical support for the theoretical results. The practical utility is illustrated in the data sets about liquor offences, claims of short-term disability and COVID-19, respectively.
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