### Peculiarities of estimating the Hurst exponent of classical Brownian motion, using the R/S Analysis

#### Abstract

The features of Hurst exponent *H* of the classical Brownian motion trajectory calculated by the R/S-analysis has been studied, where R is a range of a cumulative deviations of the chosen fragment of the trajectory within the time interval (from the mathematical aspect – time series (TS)), S is a mathematical expectation of the fragment of the analyzed TS. Due to the fact that while calculating the estimates of Hurst Exponent *H *of the analyzed TS using the R/S analysis, it is required to set up parameters values , the assumption was made on the effect of these parameters on the estimate of Hurst Exponent *H*. During the confirmation of the suggested hypothesis it was found that the estimates of the Hurst exponent *H* coincided with the accuracy up to the calculations error with the original Hurst Exponent of the classical Brownian motion equal to 0.5, only for the particular pairs of values , . It is shown that on the plane pairs of the values are located along the line When the arbitrary choice of the R/S method parameters ensures * _{k}*, , Hurst exponent varies in the span [0.25; 1.12].

The observed characteristic feature of the estimates of Hurst Exponent *H* by the R/S analysis of the classical Brownian motion makes it possible to suggest similar features of the estimates of Hurst Exponent *H* by the R/S analysis of the Fractional Brownian motion, if the suggestion proves to be true, it will be necessary to conduct a critical analysis of the results of a great number of publications where the authors used an R/S method.

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