A test for Gaussianity in Hilbert spaces via the empirical characteristic functional
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Let X_1,X_2, ... be independent and identically distributed random elements taking values in a separable Hilbert space H. With applications for functional data in mind, H may be regarded as a space of square-integrable functions, defined on a compact interval. We propose and study a novel test of the hypothesis H_0 that X_1 has some unspecified non-degenerate Gaussian distribution. The test statistic T_n=T_n(X_1,...,X_n) is based on a measure of deviation between the empirical characteristic functional of X_1,...,X_n and the characteristic functional of a suitable Gaussian random element of H. We derive the asymptotic distribution of T_n as n →∞ under H_0 and provide a consistent bootstrap approximation thereof. Moreover, we obtain an almost sure limit of T_n as well as a normal limit distribution of T_n under alternatives to Gaussianity. Simulations show that the new test is competitive with respect to the hitherto few competitors available.
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