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Let $f$ be a continous deterministic function defined on $[0,c]$ and $(B^{H}_{t})_{t\geq 0}$ be a fBM with $H\in(0,1)$. We define a Process $ (X_{t})_{t\geq 0}$ with $$X_{t}=\int_{0}^{c}f(s)B^{H}_{t+s}ds.$$ Am I right, assuming, that $(X_{t})$ is centered, gaussian and stationary?

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I have used Fubini to show that it is centered. For the normality I have tried to argue using a Riemann sum approximation. And for the stationarity I have tried to use $X_{t}=\int_{0}^{c}f(s)B_{t+s}^{H}ds=\int_{t}^{t+c}f(s-t)B_{s}^{H}ds=\int_{0}^{c‌​}f(s)(B_{t+s}^{H}-B_{t}^{H}) ds$. (substitution and stationarity of $B^{H}$. –  Peter Moor Sep 10 '12 at 21:25
    
Correction: substitution and stationary increments of $B^{H}$. –  Peter Moor Sep 11 '12 at 2:29

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