Hoelder-exponent-MFDFA-based test for long-range correlations in pseudorandom sequences

Abstract

We discuss the problem for detecting long-range correlations in sequences of values obtained by generators of pseudo-random numbers. The basic idea is that the H{ö}lder exponent for a sufficiently long sequence of uncorrelated random numbers has the fixed value $h=1/2$. The presence of long-range correlations leads to deviation from this value. We calculate the H{ö}lder exponent by the method of multifractal detrended fluctuation analysis (MFDFA). We discuss frequently used tests for randomness, finite sample properties of the MFDFA, and the conditions for a correct application of the method. We observe that the fluctuation function $F_{q}$ used in the MFDFA reacts to trends caused by low periodicity presented in the pseudo-random number generator. In order to select appropriate generators from the numerous programs we propose a test for the ensemble properties of the generated pseudo-random sequences with respect to their robustness against presence of long-range correlations, and a selection rule which orders the generators that pass the test. Selecting generators that successfully pass the ensemble test and have good performance with respect to the selection rule is not enough. For the selected generator we have to choose appropriate pseudo-random sequences for the length of the sequence required by the solved problem. This choice is based on the closeness of the H{ö}lder exponent of the generated sequence to its value 1/2 characteristic for the case of absence of correlations.