Let $p$ be an arbitrary distribution over $\mathbb{N}$, and $m\geq 1$ be an integer. Given an infinite sequence of i.i.d. draws $(X_i)_{i\geq 1}$ from $p$, define a *collision* as a pair $(i,j)$ with $i<j$ and $X_i=X_j$.

Let $M_m$ be the minimum integer $\ell$ such that $m$ collisions happen in $(X_i)_{1\leq i\leq \ell}$. I am interested in (bounds on) the expectation $\mathbb{E}[M_m]$ (as a function of $m$ and $p$; most likely its $\ell_2$-norm $\lVert p\rVert_2$).

For $p$ being the uniform distribution over $\{1,\dots,n\}$ and $m\geq 2$, this is the standard birthday paradox, and we have $\mathbb{E}[M_1] \sim_{n\to\infty} \sqrt{\frac{\pi n}{2}}$.

For $p$ being an arbitrary distribution, the (exact, although not necessarily easy to use) distribution of $M_1$ can be found in (2) of [CP00].

[CP00] also studies the general case of $M_m$ to pinpoint its exact distribution, and its limiting distribution under some assumption on $p$ (where $p$ is seen as a sequence $p=(p_n)_{n\in\mathbb{N}}$ and one lets $n\to\infty$). However, I can't seem to find how to derive simple bounds for $\mathbb{E}[M_m]$ and even $\mathbb{E}[M_1]$ from there. (By simple, I mean as above: while not necessarily tight, closed-form and depending on $m$ and simple functionals of $p$ only such as it's $\ell_r$ norms).

[CP00] Camarri, Michael, and Jim Pitman. "Limit distributions and random trees derived from the birthday problem with unequal probabilities." Electron. J. Probab 5.2 (2000): 1-18.