Limit theorem for a smoothed version of the spectral test for testing the equiprobability of a binary sequence

We consider the problem of testing the hypothesis that the tested sequence is a sequence of independent random variables that take values $1$ and $-1$ with equal probability. To solve this problem, the Discrete Fourier Transform (spectral) test of the NIST package uses the statistic $T_{Fourier}$, the exact limiting distribution of which is unknown. In this paper a new statistic is proposed and its limiting distribution is established. This new statistic is a slight modification of $T_{Fourier}$. A hypothesis about the limit distribution of $T_{Fourier}$ is formulated, which is confirmed by numerical experiments presented by Pareschi F., Rovatti R. and Setti G.