Let $X_1,X_2,\dots,X_k$ be random variables with distributions of $Bin(n_1,p_1),Bin(n_2,p_2),\dots,Bin(n_k,p_k)$ respectively.
Let us define $S = \sum_{i=1}^k X_i$
We know that if $\ p_i =p_j \ \ \ \forall i,j$, and that all of the random variables are independent, we get that $S \sim Bin(\sum_{i=1}^k n_i, p)$. (where $p=p_i$ for some $i$)
My question is this: Are there any other conditions (weaker?) when we don't demand that the probabilities are equal or that they are independent (maybe we add a different assumption) where we still get $S \sim Bin(m,q)$ for some values $q,m$ and if so what are they, and what are the values $q$ and $m$?
Thanks!