A random-projection based test of Gaussianity for stationary processes

Gaussianity tests have being widely studied in the literature. Regarding the study of Gaussianity tests for stationary processes, these only verify the Gaussianity of a marginal at a fixed finite order, generally order one. Therefore, they do not reject stationary non-Gaussian processes with the one-dimensional Gaussian marginal. Thus, a consistent test is proposed for Gaussianity of stationary processes when a finite sample path of the process is observed. Using random projections, decision rules are applied to the whole distribution of the process and not only on its marginal distribution at a fixed order, as in previous tests. The main idea is to test the Gaussianity of the one-dimensional marginal distribution of some random linear transformations of the process. Note that testing the one-dimensional marginal distribution can be done with previous tests of Gaussianity for stationary processes. It is shown by both theoretical and empirical studies that the proposed test procedure has good properties for a wide range of alternatives.