Second order properties of nearly nonstationary ARMA processes are investigated in the cases when the autoregressive polynomial equation has (i) a real root close to 1; (ii) a real root close to −1; (iii) a pair of complex roots close to the unit circle.

The effect of the closeness to the unit circle of the ARMA poles on its covariance and spectral density functions is considered. The obtained results demonstrate three specific ways of degeneracy of these functions, as the roots tend to 1 in modulus. As a consequence three different estimates of the ARMA parameters located in the neighbourhood of the border of the stationarity region for ARMA process are derived and their asymptotic distributions are examined.