Tests for the Gaussianity of a time series with application to financial time series

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In a view of testing multivariate normality, a new testing procedure for testing the Gaussianity of stationary time series process is proposed. The basic idea is that first standardizing p variates and then transforming them by Hermitian polynomials, the moment structure of the p-dim. multivariate normality is tested based on the transformed variates. The test statistic is shown to be asymptotically distributed as x^2-distribution. These results are applied to testing the Gaussianity of a univariate time series by considering lagged variates p-dim variates. The method is applied to returns of stocks and the Gaussianity is rejected.Also the problem of testing multivariate normality against a curved family is treated and the locally best invariant test and its distribution are derived. A curved family for which the Mardia test is locally best invariant is found.