pr.probability - Problem with a Long Range Correlated Time Series

Consider a stochastic process $X_t$ , $t in 1,2,3,..,N$.

$X_t$ is a Bernoulli variable and $Pr(X_t=1) = p$ for all $t$.
The Autocovariance function $gamma(|s-t|)= E[(X_t - p)(X_s -p)]$ is given

$gamma(k) = frac{1}{2} (|k-1|^{2H} - 2|k|^{2H} + |k+1|^{2H}).$

For a constant $Hin (0,1)$ This is the same autocovariance as for fractional gaussian noise (increments of the fractional brownian motion), and give a autocovariance which falls like a power law when $k$ goes to infinity.

Let X and Y be process with the given properties, I am interested in the following probability distribution:

$Prleft(sum_{i=0}^N X_i Y_i = kright)$

That is the distribution of the overlap of two such processes. For $H=1/2$ the process is not correlated and I have the simple result that $Pr(X_t Y_t)=p^2$, and that

$Prleft(sum_{i=0}^N X_i Y_i = kright) = {N choose k} p^{2k} (1-p^2)^{N-k}.$

But for $Hneq 1/2$, I do not know how to deal with the long range correlation. Is there a way to proceed on this problem? I regret i never took a class in Stochastic Analysis, and I really hope the question makes sense. Any help or input would be highly appreciated.

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