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 $H\in (0,1)$ 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:


$
\Pr\left(\sum\_{i=0}^N X\_i Y\_i = k\right)
$


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


$
\Pr\left(\sum\_{i=0}^N X\_i Y\_i = k\right) = {N \choose k} p^{2k} (1-p^2)^{N-k}.
$


But for $H\neq 1/2$, 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.



Problem with a Long Range Correlated Time Series
One thing you should understand is that Bernoulli is not Gaussian: the autocorrelation function does not determine the process uniquely. In particular, the fact that the Bernoulli variables are not correlated doesn't mean that they are independent. For instance, the 3 step process that takes the paths (0,0,0),(0,1,1),(1,1,0),(1,0,1) with probability $1/4$ each has no autocorrelations but $\sum\_{i=0}^2 X\_iY\_i$ is never $3$ here. So, your formula fails for this process. We need to know much more than just autocorrelations to answer your question.


