Let $(X_1,X_2,\ldots,X_k)$ be a multinomial random vector with parameters $n, p_1, p_2, \ldots, p_k$ (i.e., we throw randomly $n$ balls into $k$ bins, so that for each ball, the probability of landing in bin $i$ is $p_i$, independent of other balls).

Let $Y = \sum_{i=1}^k I_{\lbrace X_i = 0\rbrace }$ be the number of empty bins.

How is $Y$ distributed? Will be grateful to learn about any exact/approximate/asymptotic result.

I am familiar with the "occupancy problem" of Feller (the special case in which $p_i = 1/k$ for all $i$), and also with the Bahadur representation for dependent binary variates.