A stationary sequence $(X_n)_{n\in\mathbb{N}}$ exhibits long-range dependence if the autocovariance function $\rho(n):=\mathrm{cov}(X_k,X_{k+n})$ satisfy $\lim\limits_{n\to\infty}{\rho(n) \over cn^{-\alpha}}=1$ for some constant $c$ and $\alpha\in (0,1)$.
The increments of the fractional Brownian motion have the long-range dependence property for $H>{1\over2}$ since $\rho_H(n)={1 \over2}\left[ (n+1)^{2H}+(n-1)^{2H}-2n^{2H}\right]\sim H(2H-1)n^{2H-2}$ as $n$ goes to infinity.
I cannot follow why this relation holds. Can anybody help please?