Geodesic
Limit theorem for a smoothed version of the spectral test for testing the equiprobability of a binary sequence
Diskretnaya Matematika, Tome 33 (2021) no. 4, pp. 132-140

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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.
Keywords: discrete Fourier transform test, spectral test, NIST, TestU01, Rademacher distribution } The author is grateful to A.M. Zubkov for constant attention. {\begin{thebibliography}{9. 
@article{DM_2021_33_4_a10,
     author = {M. P. Savelov},
     title = {Limit theorem for a smoothed version of the spectral test for testing the equiprobability of a binary sequence},
     journal = {Diskretnaya Matematika},
     pages = {132--140},
     publisher = {mathdoc},
     volume = {33},
     number = {4},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2021_33_4_a10/}
}
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M. P. Savelov. Limit theorem for a smoothed version of the spectral test for testing the equiprobability of a binary sequence. Diskretnaya Matematika, Tome 33 (2021) no. 4, pp. 132-140. http://geodesic.mathdoc.fr/item/DM_2021_33_4_a10/