Is the following equality right? Let $(B^H)$ be the fBM $ \mathbb{E}\left[\sum_{k=1}^n\alpha_k\left(B^H_{\frac{k}{c_n}}-B^H_{\frac{k-1}{c_n}}\right)\cdot\sum_{k=1}^n\alpha_k\left(B^H_{\frac{x+k}{c_n}}-B^H_{\frac{x+k-1}{c_n}}\right)\right] =c_n^{-2H}\mathbb{E}\left[\sum_{k_1,k_2=1}^n\alpha_{k_1}\alpha_{k_2}X_{k_1}X_{k_2}\right] $ Where $c_n>n$ is integer valued and $x\in\mathbb{R}$ and >0 and $(X_\ell)_{\ell\geq 0}$ is the fractional Gaussian noise.
Equality of Expectations
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probability
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0If you want to see if some cancellations occur, write down the sum and check. For that, you should write down explicitly $\gamma$. – 2012-11-06