let X = {x_i}, and Y = {y_i} be two vectors of binary variables with n elements (taking the value 0 or 1 with some probability). X and Y have the same distribution.

I'm interested in the following quantity

Pr( Sum_{i=1}^n x_i *y_i )

That is, the probability distribution the 'overlap' of the two vectors (sum from i=0 to n). I have managed to find the distribution for this if all the x_i are not correlated. Its just a simple binomial. But want I really to calculate is the probability dist. of the overlap when the x_i (and y_i) are correlated in the following way.

E[ x_i x_j ] = E[ y_i y_j ] = const * |i-j|^-alfa

where alfa is a positive number.

Is this possible? As I do not have a background in probability theory, I hope the question make sense. I would highly appreciate any help or suggestions.

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jon