The procedures known until now to test if a given stationary process is Gaussian determine when the one-dimensional marginals of the stationary process are Gaussian. Obviously, these provide tests at the right level for the intended problem; but these tests could be at the nominal power against some non-gaussian alternatives as those stationary non-Gaussian process with Gaussian marginals. The procedure we propose to test if a given stationary process of real-valued random variables, X=(X_n) is Gaussian is a combination of the random projection method joined to some available procedures which allow to determine when the one-dimensional marginals of a stationary process are Gaussian. Here, we have chosen the proposal by Epps.

So, we are interested in constructing a test for the null hypothesis H_0: "X is Gaussian". Notice that H_0 holds if and only if (X_1,..., X_n) is Gaussian and, using the random projection method, this is, roughly speaking, equivalent to that the projection of (X_1,..., X_n) in random Hilbert-valued element is Gaussian. Now, is the turn of using Epps's procedure which checks if the characteristic function of the one-dimensional marginal of a stationary process coincides with the characteristic function of a Gaussian distribution in a finite set of points. The difference here is that the points employed in Epps's procedure are also chosen at random what guarantees the consistency of the whole test.

This is a joint work with Juan Antonio Cuesta-Albertos, Universidad de Cantabria, and Fabrice Gamboa, Université de Toulouse.