The author deals with the autoregressive model where is the -th observation on the dependent variable, for , is the characteristic polynomial, is the backward shift operator, i.e., , and the disturbance process is given by , where is a sequence of i.i.d. random variables, is a sequence which decays hyperbolically, i.e., , , , and is a slowly varying function, bounded on every finite interval. The unknown parameter is estimated by the least squares estimate (LSE):
where . The least squares error satisfies
The asymptotic distribution is derived for the least squares estimates (LSE) only in the case when the characteristic polynomial is unstable with complex-conjugate unit roots, i.e., an appropriate non-stationary model to identify persistent cycles in non-Gaussian long-memory time series. The behaviour of the LSE when has stable roots, roots equal to -1 and 1, and explosive roots remains an open problem. In order to describe the limiting distribution of the LSE, two limit theorems involving long-memory processes are proved. The first theorem gives the limiting distribution of the weighted sum, , where is a non-Gaussian long-memory moving-average process and is a given sequence of weights. The second theorem is a functional central limit theorem for the sine and cosine Fourier transforms , , where and .