I am taking some samples from multinomially identically distributed random variables $X^i$ :

$X^i = (X^i_1, ..., X^i_m) \sim Multinomial(n, p_1,...,p_m)$

Denote the vector: $X^* = \sum_i X^i$, i.e. a sample from $X^*$ is the sum of i.i.d. multinomial samples.

Now, how would the PDF/CDF of the distribution of $X^*$ would look like? Is it possible to express it in a closed form?

An example of what I would like to compute with that is as follows: Denote the event $A = X^*_2 \ge 1 \land X^*_3 \ge 1 \land ... \land X^*_m\ge1$.

- What is $P(A)$ for a given $n$?
- What is the smallest $n$ such that $P(A) > \alpha$?

If that makes it any easier, it would also be interesting to know the answer to a special case when $p_2 = p_3 = ... = p_m$.

I am thinking that perhaps the distribution of $X^*$ itself is also multinomial, but I am not sure how to prove that.