Let $ X $ to be a random variable taking values in $ {0,1 } ^ n $ with the next distribution. For each coordinate $ i $, we have $ p_i = P (X_i = 1) = c / sqrt n $, or $ c $ is a (very small) constant. Coordinates $ i $ and $ j have positive correlations decreasing exponentially in $ | i-j | $, with prefactor 1 $ / sqrt n $, in the following sense: writing $ p_ {i, j} = P (X_i = 1, , X_j = 1) $, we have

$$ 0 the p_ {i, j} – p_i p_j the exp (- | i-j |) / sqrt n. $$

It's a *dependent Bernoulli process*.

Let $ mu $ denotes the law of that.

Also, leave $ Y $ bean *independent Bernoulli process* with the same marginals: $ P (Y_i = 1) = p_i $ and the coordinates are independent.

Let $ nu $ denotes the law of that.

I want to limit the total variation distance $ | mu- nu | _ text {TV} $.

In particular, I want to show that the television distance decreases with $ c $, that is, by taking $ c to $ 0 given $ text {TV} to 0 $.

I know the Chen-Stein method to address such questions, but it seems to me more appropriate when the probabilities $ p_i $ are in order $ 1 / n, and so there is a number of $ 1 $s in the independent case (and the method shows that it is the same for dependent cases, under certain conditions). Perhaps Stein's method can be applied more generally?

In addition, the purpose of this question is not to obtain an accurate answer from anyone, but rather a reference or suggested approach method. The above is a simplified version of my current problem, but I feel that if I can understand the above, I will be able to convert it to my specific case.