Hi,


let $\gamma(k) = 1/2 (|k+1|^{2H} + |k-1|^{2H}-2|k|^{2H}),k\in\mathbb{Z},$ be autocovariance function of fractional Gaussian noise where $H\in(0,1)$ is parameter.


I want to show that $\gamma$ is strictly positive definite function, i. e., for $n\in\mathbb{N}$ the matrix $\Gamma = (\gamma(i-j))\_{i,j=1}^n$ is (strictly) positive definite.


I tried to do that by using tools of linear algebra, unfortunately I wasn't successful. I spent some time googling and found that Bochner's theorem [1] might help me. I don't understand how and I wasn't able to find an explanation of the application either.


Is it possible to show it using Bochner's theorem? Or is it possible to do it in a different way?


Thank you very much!


Tomas


[1] <http://en.wikipedia.org/wiki/Bochner%27s_theorem#Applications>



Strictly positive definite autocovariance function of fGn
As per Mathematica, the inverse Fourier transform of your function $\gamma(x)$ is


\begin{equation\*}
\frac{\sqrt{2 \pi } \sec \left(\frac{\pi h}{2}\right) (\cos (t)-1) |t|^{-h-1}}{\Gamma (-h)}
\end{equation\*}


But for $0 < h < 1$, $\sec(\pi h/2) / \Gamma(-h)$ is negative, also $\cos(t)-1 \le 0$, so overall the IFT is nonnegative. Thus, $k(x,y) := \gamma(x-y)$ is a positive definite function. For strict positive definiteness some more work might be needed.


