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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0Independently of $x$? – 2012-11-06
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0by stationarity of increments? – 2012-11-06
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0I mean, the LHS seems to depend on $x$, not the RHS, hence are you sure this is the statement you want to prove? – 2012-11-06
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0I want to calculate the LHS. I thought, that it does not depent on on $x$ because of stationarity. – 2012-11-06
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0can anyone give me a feedback?Again I want to calculate the LHS. Thanks in advance! – 2012-11-06
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0Since $(B^{H}_{x+k}-B^{H}_{x+k-1})\overset{d}{=}(B^{H}_{k}-B^{H}_{k-1})$ no $x$ appears on the RHS. Is there a fundamental mistake? – 2012-11-06
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0Yes. Which is that one should multiply each of these by the increments in the first sum--and then the dependence structure enters the scene. – 2012-11-06
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0Let $X_{k_{1}}:=B^{H}_{k_{1}}-B^{H}_{k_{1}-1}$ and $Y_{k_{2}}:=B^{H}_{x+k_{2}}-B^{H}_{x+k_{2}-1}$. So the LHS is $\sum_{k_{1},k_{2}=1}^{n}\alpha_{k_{1}}\alpha_{k_{2}}\mathbb{E}\left[X_{k_{1}}Y_{k_{2}}\right]$ Using the correlation structure of the fBM we get: $\mathbb{E}\left[X_{k_{1}}Y_{k_{2}}\right]=\frac{1}{2}\gamma(k_{1},x+k_{2})-\frac{1}{2}\gamma(k_{1},x+k_{2}-1)-\frac{1}{2}\gamma(k_{1}-1,x+k_{2})+\frac{1}{2}\gamma(k_{1}-1,x+k_{2}-1)$ where $\gamma(s,t)=\mathbb{E}\left[B^{H}_{s}B^{H}_{t}\right]$ Is it possible to do further simplifications? – 2012-11-06
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0Yes. Plug in $\gamma$. – 2012-11-06
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0Ok, but after I have plugged in $\gamma$ in the sum, there is no way to simplify the sum? Iam sorry because of the typos in my above comment, but I have tried to fix them $\infty$ many times, but it still does not work. – 2012-11-06
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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