I'm trying to investigate asymptotic behavior of this sum:
$$\lim_{n->\infty}\sum_{i=1}^nX_iX_{i-1},$$
where $X_i\sim\mathcal{N}(0,1)$ and $cov(X_i, X_i)=1$, $cov(X_i, X_{i-1})=const.$
So, far I managed to find PDF for this product:
$$f_{X_iX_{i-1}}=\frac{1}{\pi\sqrt{1-\rho^2}}\exp\Bigg[\frac{\rho z}{1-\rho^2}\Bigg]K_0\Bigg(\frac{|z|}{1-\rho^2}\Bigg),$$
where $K_0$ is Bessel function, $\rho$ - correlation coeficient.
However, I don't know where to go futher from there.
