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.](http://projecteuclid.org/euclid.aop/1176991491).


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

\begin{equation\*}
W\_1 : = \sum\_{i=1}^n X\_i, \ \ \ 
W\_2 : = \sum\_{j=1}^n Y\_j.
\end{equation\*}
are sums of (not necessarily independent) Bernoulli random variables, then can we bound $d\_{TV}(\mathcal{L}(W\_1, W\_2), \mathcal{L}( Poi(E[W\_1]), Poi(E[W\_2]))$, using the first and second moments of $X\_i$ and $Y\_j$ (these poisson random variables are independent)$? In the literature, I've only found bounds for the case where these Bernoulli random variables are presumed to be independent.


EDIT: Whoops. I should have read through the paper that I cited. See Theorem 2 in Arratia et al.



Approximating by independent Poisson random variables
The short answer is "yes".


Longer answer: it is enough to show that  

the variable
$Z:= W\_1+ W\_2$ converges to Poisson with the correct parameter. Now apply the "Two moments suffice". Note that if the $W\_1$ and $W\_2$ are dependent,
the conditions of the latter may fail - but at least they can be checked using first and second (including cross) moments.


Added in edit: In case that the dependence between the X and Y variables is strong, the
limiting Poisson variables may be correlated, and the 
limit of $W\_1+W\_2$ might be compound Poisson instead of Poisson (this would be the case for example if $X\_i=Y\_i$). For the local method for compound Poisson approximation, see e.g. <http://projecteuclid.org/euclid.aoap/1177004910>


