A borderline Gaussian random Fourier series for the sample convergence in variation

We give an example of a Gaussian random Fourier series, of which the normalized remainders have their sojourn times converging in variation, namely the convergence in the space L1(R) of the related density distributions, to the Gaussian density. The convergence in the space L∞(R) is also proved. In our approach, we use local times of Gaussian random Fourier series.