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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
\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.

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  • $\begingroup$ What is $\mathcal{L}$? $\endgroup$ – Nate Eldredge Mar 15 '15 at 17:23
  • $\begingroup$ By $\mathcal{L}$, I mean the law (distribution) of these random variables. $\endgroup$ – D Poole Mar 15 '15 at 17:28
  • $\begingroup$ If $W_1, W_2$ are dependent, it seems unreasonable for the law of $(W_1, W_2)$ to be well approximated by a pair of independent Poissons. For example, what happens if $X_i = Y_i$ so that $W_1 = W_2$? $\endgroup$ – Nate Eldredge Mar 15 '15 at 17:33
  • $\begingroup$ In that case, the second (mixed) moments would hopefully be too large and give an upper bound on the total variation distance that would be meaningless. For example, in the 1-dimensional case, if there is too much dependence between the $X_i$'s, the $b_2 = \sum_{\alpha} \sum_{\beta \in N_{\alpha}} E[X_{\alpha} X_{\beta}]$ term would be large (here $N_\alpha$ can be thought of as the dependence neighborhood of $X_\alpha$). $\endgroup$ – D Poole Mar 15 '15 at 18:04
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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

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  • $\begingroup$ Many thanks for your answer. I am having trouble seeing why $W_1$ and $W_2$ would be forced to be asymptotically independent in this case rather than simply asymptotically uncorrelated. $\endgroup$ – D Poole Mar 17 '15 at 18:02

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