# Approximating dependent random variables with bounded covariance, by independent variables

Suppose we want to compute the probability of some condition on some random variables. We know how to compute it only for independent random variables. Our variables are dependent, but their covariance is small. Can we use this fact to get an approximation?

For concreteness, let $X_1,\ldots,X_n$ be normal random variables with, for all $i,j$:

• $\operatorname{Mean}(X_i) = 0$,
• $\operatorname{Var}(X_i) = v$,
• $\operatorname{Cov}(X_i,X_j) = c$.

Let $E$ be some condition on the random variables, and let $f(v,c)$ be the probability of $E$ as a function of $v$ and $c$.

Suppose we know to compute $f(v,0)$ for every $v$. I.e, we know to compute the probability of $E$ when the variables are statistically-independent. Can we use this to derive a bound on $f(v,c)$?

• If these are multivariate normal and none of the correlations are $\pm1$ then $\Pr[X_3 = X_5 + X_7] =0$ – Henry Sep 15 '17 at 20:19
• @Henry that's only an example... I am looking for a general bound – Erel Segal-Halevi Sep 17 '17 at 4:10