# Approximating by independent Poisson random variables

Using the Chen-Stein method, one can bound the total variation distance between a sum of possibly dependent Bernoulli random variables $W=\sum_{i=1}^n X_i$ and a Poisson distribution using only the first and second moments of the $X_i$. For example, see Two Moments Suffice for Poisson Approximations, by Arratia et al..

Is there an analogous bound for the total variation distance in the multivariate case? For example, if