Estimating the Hurst Exponent via Rescaled Range Analysis

You observe a time series $x_1, x_2, \ldots, x_T$ and want to determine whether it exhibits mean-reversion ($H < 0.5$), random-walk behavior ($H = 0.5$), or persistence ($H > 0.5$). 1. Define the rescaled range statistic $R/S$ for a block of $n$ observations. Show how the scaling relationship $E[R/S] \sim c \, n^H$ gives you the Hurst exponent $H$. 2. Provide a step-by-step algorithm to estimate $H$ from data. Be specific about how you choose block sizes, how you compute $R/S$ for each block, and how you extract $H$ from the log-log regression. 3. Give two concrete reasons -- with mathematical intuition -- why heavy tails or volatility clustering can bias $\hat{H}$ upward, making an uncorrelated series look persistent.