Call $(X_k)_{k\geqslant0}$ the values of the process observed at time intervals of length $T$. Ergodic estimators of $H$ based on the second moment are $$ \widehat H_n=\frac{\log\left(\sum\limits_{k=1}^n(X_k-X_{k-1})^2\right)-\log n}{2\log T}. $$ For a study of $\widehat H_n$ and some comparisons to other approaches, see New Estimation Techniques for Fractional Brownian Motion by V. Dobric, D. Scansaroli, R. Storer.