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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?

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