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I hope someone can help me with this:

Let $c(\rho)=\lim_{n\rightarrow\infty}n^{\alpha-1}\sum_{k=-(n-1)}^{n-1}\left[1-\frac{k}{n}\rho(k)\right]$ where $n=1,2,3,\ldots$, $\alpha\in(0,1)$ and $\rho$ is a correlation function which depends only on the lag $k$ $(-1\leq\rho(k)\leq 1)$.

I want to show that with increasing $n$,

$\sum_{-(n-1)}^{n-1}\rho(k)\approx constant \times n^{1-\alpha}$ $\ \ \ \ \ (1)$

and to conclude that, as $\alpha$ is less than $1$,

$\sum_{-\infty}^{\infty}\rho(k)=\infty$ $\ \ \ \ \ (2)$

that is, the correlations are not summable. Finnaly show that, relation (1) holds if $\rho(k)\approx c_{\rho}|k|^{-\alpha}$, as $k$ goes to infinity and $c_{\rho}$ is finite positive constant.

Thanks!!

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