# How to approximate the distribution of the sum of multiple multinomial random variables?

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?

• 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. – whuber Sep 20 '19 at 18:46
• 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. – Glen_b -Reinstate Monica Sep 22 '19 at 4:01

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.