Moment Approach for Quantitative Evaluation of Randomness Based on RMT Formula
We develop in this article a quantitative formulation of the randomness-test based on the random matrix theory (RMT-test), in order to compare a subtle difference of randomness between given random sequences. Namely, we compare the moments of the actual eigenvalue distribution to the corresponding theoretical expression that we derive from the formula theoretically derived by the random matrix theory. We employ the moment analysis in order to compare the eigen-value distribution of the cross correlation matrix between pairs of sequences. Using this method, we compare the randomness of five kinds of random data generated by two pseudo-random generators (LCG and MT) and three physical generators. Although the randomness of the individual sequence can be quantified in a precise manner using this method, we found that the measured values of randomness fluctuate significantly. Taking the average over 100 independent samples each, we conclude that the randomness of the random data generated by the five generators are indistinguishable by the proposed method, while the same method can detect the randomness of the derivatives of the sequences, or the initial part of LCG, which are distinctly lower.
KeywordsRandomness measure RMT-test Moment analysis Eigenvalues of cross correlation matrix Marcenko-Pastur distribution
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