Consider the classic balls in bins problem: we throw balls one by one into $n$ bins independently and uniformly. Define $\tau(k)$ for $1 \le k \le n$ to be the number of balls we have thrown until $k$ bins are occupied, i.e., have at least one ball inside.
I am looking for a reasonable $t$ such that $\mathbb{P}(\tau(k) \ge t) = o(n^{-1})$. That is, the probability $\mathbb{P}(\tau(k) \ge t)$ goes to $0$ faster than $1/n$ when $n \to +\infty$.
Here $t$ can be some value depending on $k$ and $n$. Of course we hope $t$ to be as small as possible.
This can also be a variation of the classic coupon collector problem: let $\tau(k)$ be the number of trials we have made until $k$ distinct coupons are collected.
I have searched for this problem for a while. It seems the primary interest on this model is the maximum load and the expected number of empty bins left after $n$ throws. I haven't seen any related work on the problem I am interested in.
Thank you.