Let $X$ be a random variable taking values in $\{0,1\}^n$ with the following distribution. For each coordinate $i$, we have $p_i = P(X_i = 1) = c/\sqrt n$, where $c$ is a (very small) constant. Coordinates $i$ and $j$ have positive correlations exponentially decaying 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 $$ 0 \le p_{i,j} - p_i p_j \le c \exp(-|i-j|)/\sqrt n. $$ This is a dependent Bernoulli process. Let $\mu$ denote the law of this.

Also, let $Y$ be an independent Bernoulli process with the same marginals: $P(Y_i = 1) = p_i$ and coordinates are independent. Let $\nu$ denote the law of this.

I want to bound the total variation distance $\|\mu-\nu\|_\text{TV}$. In particular, I want to show that the TV distance decays with $c$, ie taking $c \to 0$ gives $\text{TV} \to 0$.

I am aware of the Chen-Stein method for approaching questions like this, but to me this seems better suited when the probabilities $p_i$ are order $1/n$, and so there are a Poisson number of $1$s in the independent case (and the method shows that the same holds for dependent case, under certain conditions). Perhaps one can apply Stein's method more generally?

Also, the aim of this question isn't to get a precise answer from someone, but rather a reference or suggested method of approach. The above is a simplified version of my actual problem, but I feel that if I can get a good handle on the above, then I can convert it to my specific case.


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