ИСТИНА

We consider the AR$(q)$ model \[ X_n=\beta^{(1)}X_{n-1}+\dots+\beta^{(q)}X_{n-q}+\varepsilon_n, \] where $\varepsilon_n$, $n\ge 1$, are independent identically distributed random variables with $\EE \varepsilon_n=0$ and $\EE \varepsilon_n^2 < \infty$. It is assumed that the model is \emph{unstable}, which means that the characteristic polynomial \[ \phi(z) = 1 - \beta^{(1)}z - \dots - \beta^{(q)}z^q \] has at least one root on the unit circle and has no roots inside the unit circle. Jeganathan (1995) studied the limit behaviour of likelihoods in this model under the additional assumption that $\varepsilon_n$ have the density $p(x)$ with a finite Fisher information. Our purpose is to investigate the same problem in the ``almost regular'' case: \[ \int\bigl(p^{1/2}(x)-p^{1/2}(x+t)\bigr)^2\,dx = t^2 l(t), \] where $l(t)$ varies slowly and $l(t)\to\infty$ as $t\to 0$.