Abstract:

Hurst parameter is widely used to characterize long memory and self-similarity in series. This paper introduces the relations between the Hurst parameter originally calculated by the $R/S$ statistic and the self-similarity, long memory, heavy tails under finite second moment and under heavy tails. In the case of finite second moment, the Hurst parameter calculated by the $R/S$ statistic is consistent with the self-similar parameter, and can describe the long memory defined by covariance. In the case of heavy tails with infinite second moment, the covariance linking Hurst parameter with long memory is infinite, and their relationship cannot be discussed. The $R/S $statistic is not necessarily related to self-similar parameters and tail index either. These contents can make the practical meaning of Hurst parameter clear.

Key words: Hurst parameter, Long memory, Heavy tails, Self-similarity, Fractional Brownian motion