Symmetry verification of innovation distributions in autoregressive schemes

We consider a stationary linear $AR(p)$ model with zero mean. The autoregression parameters as well as the distribution function (d.f.) $G(x)$ of innovations are unknown. We test symmetry of innovations with respect to zero in two situations. In the first case the observations are a sample from a stationary solution of $AR(p)$. We estimate parameters, find residuals. Based on them we construct a kind of emperical d.f. and the omega-square type test statistic. Its asymptotic d.f. under the hypothesis and the local alternatives are found. In the second situation the observations subject to gross errors (outliers). For testing the symmetry of innovations again we construct the Pearson's type statistic and find its asymptotic d.f. under the hypothesis and the local alternatives. We establish the asymptotic robustness of Pearson's test as well.