I'm looking for a generalization to the urn-ball matching problem. As a reminder of what I've got in mind, here's the simple version:

Randomly assign (with replacement) $N$ balls to $M$ urns. Afterwards, for any urn with more than one ball assigned to it, extract one at random.

The total number of matches (urn-ball pairs) will be $M(1-(1-\frac{1}{M})^N)$ in expectation. To see this, fix an urn. The probability of this urn being matched is the probability that at least one ball arrives. Since the probability that no ball arrives is $(\frac{M-1}{M})^n$, we get the expected number of matches above. From a given ball's perspective, the probability of being matched is $\frac{M}{N}(1-(1-\frac{1}{M})^N)$.

We care what happens when there is a measure $N$ of balls and a measure $M$ of urns, so we take the limit of these expressions as $\theta=\frac{M}{N}\rightarrow \infty$ and get the matching function $\mathcal{M}(M,N)=M(1-e^{-\frac{M}{N}})$ and the ball's matching probability $\frac{\mathcal{M}(M,N)}{N}:=p(\theta)=\theta(1-e^{-\frac{1}{\theta}})$.

We generalize this in the following way: introducing heterogeneity in the sizes (weights) of the balls, which leaves static the assignment probability for a given urn but changes the probability of extraction--now, instead of randomly extracting a ball from an urn, we extract each ball with probability equal to its relative weight within the urn.

To fix ideas, let $n_i$ be the number of balls of size $U_i$ in the "population" of balls, and let $N$ be the number of types of ball ($M$ is still the number of urns).

With a small bit of effort it's not hard to see that the probability of a given ball of size $U_i$ being extracted is given by:

$P_i (\textbf{n},\textbf{U};M,N) =\sum \limits_{k_1 = \delta_{i,1}} ^{n_1}\cdots\sum \limits_{k_N=\delta_{i,N}} ^{n_N}\prod \limits_{j=1}^N \dbinom{n_j-\delta_{i,j}}{k_j-\delta_{i,j}}\frac{M(M-1)^{\sum \limits_{j=1}^N (n_j-k_j)}}{M^{\sum \limits_{j=1}^{N} n_j}}\frac{U_i}{\sum \limits_{j=1}^N k_j U_j}$

Where $\delta_{i,j}$ is Dirichlet: 1 if $i=j$, 0 otherwise, $i \in \{1,\ldots,N\}$.

To see why, note that any possible arrangement in a given urn is fully characterized by the total number of each type of ball present. Thus this expression sums over the possible total numbers of each ball in a given urn--these are the indices $k_j$, which range over $\{0,\ldots,n_j\}$ except when $j=i$, because we know there is at least one ball of type $i$.

The term $\prod \limits_{j=1}^N \dbinom{n_j-\delta_{i,j}}{k_j-\delta_{i,j}}M(M-1)^{\sum \limits_{j=1}^N (n_j-k_j)}$ multiplies by the number of urns $M$ the number of ways to choose $k_j$ balls from the $n_j$ of each type (and $k_i-1$ from $n_i-1$ *other* type $i$ balls) multiplied by the number of ways to distribute the remaining $n_j-k_j$ balls to the other $M-1$ urns. The denominator $M^{\sum \limits_{j=1}^{N} n_j}$ is the total number of possible assignments of the balls to the urns.

Lastly the term $\frac{U_i}{\sum \limits_{j=1}^N k_j U_j}$ gives the probability of $U_i$ being taken from the urn with $k_j$ of each type of ball.

My problem concerns finding something tractable here (as above) for what happens in the limit when we have a measure $M$ of urns and a measure $\eta = \sum \limits_{j=1}^N n_j$ of balls, sending $N\rightarrow \infty$ keeping $\frac{M}{\eta}$ fixed. In the limit there will be a density function associated with each ball size $f_U$ which gives the relative presence of balls of size $U$ for all $U$ in its support. So that denominator should become an integral like $\intop f_U(u) du$

Also note that: $\sum \limits_{k_1 = \delta_{i,1}} ^{n_1}\cdots\sum \limits_{k_N=\delta_{i,N}} ^{n_N}\prod \limits_{j=1}^N \dbinom{n_j-\delta_{i,j}}{k_j-\delta_{i,j}}\frac{M(M-1)^{\sum \limits_{j=1}^N (n_j-k_j)}}{M^{\sum \limits_{j=1}^{N} n_j}}=1$ (which can be seen via repeated application of the binomial theorem quite quickly once you see how the $N=1$ case works).

Any ideas on how to proceed would be awesome!! I'm not facile enough with the binomial coefficients to simplify this expression to the point where I could take a limit.