  We consider a stationary linear AR($p$) model with unknown mean. The
autoregression parameters as well as the distribution function (d.f.) $G$ of
innovations are unknown. The observations contain gross errors (outliers). The
distribution of outliers is unknown and arbitrary, their intensity is $\gamma
n^{-1/2}$ with an unknown $\gamma$, $n$ is the sample size. The assential
problem in such situation is to test the normality of innovations. Normality,
as is known, ensures the optimality properties of widely used least squares
procedures. To construct and study a Pearson chi-square type test for normality
we estimate the unknown mean and the autoregression parameters. Then, using the
estimates, we find the residuals in the autoregression. Based on them, we
construct a kind of empirical distribution function (r.e.d.f.) , which is a
counterpart of the (inaccessible) e.d.f. of the autoregression innovations. Our
Pearson's satatistic is the functional from r.e.d.f. Its asymptotic
distributions under the hypothesis and the local alternatives are determined by
the asymptotic behavior of r.e.d.f. In the present work, we find and
substantiate in details the stochastic expansions of the r.e.d.f. in two
situations. In the first one d.f. $ G (x) $ of innovations does not depend on $
n $. We need this result to investigate test statistic under the hypothesis. In
the second situation $ G (x) $ depends on $ n $ and has the form of a mixture $
G (x) = A_n (x) = (1-n ^ {- 1/2}) G_0 (x) + n ^ { -1/2} H (x). $ We need this
result to study the power of test under the local alternatives.
