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Say we have $T$ independent Multinomial random variables $X_1,X_2\dots X_T$, with $p(X_t=m)=p_{t,m},m\in\{1,2,...M\}$. What would be the distribution of $X_1+X_2+...+X_T$? If there is no closed-form, is there any approximation version of this distribution? Like, is there a multivariate version of possion-binomial distribution?

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    $\begingroup$ Could you explain the sense in which $X_1+\cdots + X_N$ could be considered "multivariate"? One wonders whether you use the term "Multinomial random variables" in the standard sense of, say, Wikipedia. $\endgroup$ – whuber Sep 20 '19 at 18:46
  • $\begingroup$ Your question is ambiguous/unclear. If you're summing the counts in a subset of the categories for a single multinomial, the sum will be binomial. $\endgroup$ – Glen_b -Reinstate Monica Sep 22 '19 at 4:01
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Assuming you mean multinomial distribution in the usual sense (in which case you should correct the range to include zero.) Then: If the probability parameter $p=(p_1, \dotsc, p_k)$ are all equal, then the sum is also multinomial. If they are unequal, then we have the situation that in the binomial $k=2$ case leads to Poisson-Binomial distribution, and in the multinomial case is known as (surprise!) Poisson-Multinomial. Here is a relevant math SE post.

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