Limit Theory of the Multi-set Allocation Occupancy (MAO) Distribution: Normal and Poisson Approximations via MAO Norm

This paper investigates the asymptotic behavior of the Multi-set Allocation Occupancy (MAO) distribution, which models the count vector $X=(X_{=0},\dots ,X_{=T})$ from $T$ independent rounds of sampling without replacement of size $m$ from $N$ individuals. Focusing on $X_{=t}$ (individuals in exactly $t$ subsets) and employing the MAO norm – a combinatorial tool yielding closed-form factorial moments – we derive the exact marginal distribution of a single individual as $\mathrm{B}\mathrm{i}\mathrm{n}(T,p)$ with $p=m\mathrm{/}N$. Using the MAO norm, we prove that for any fixed number of distinct individuals, their joint distribution differs from the product of marginals by $O(1\mathrm{/}N)$, establishing the weak dependence required for limit theorems. Based on these findings, we delineate two asymptotic regimes: 1. Normal approximation: When $N\to \mathrm{\infty }$ with $p=m\mathrm{/}N$ fixed, $X_{=t}$ obeys a central limit theorem and can be approximated by a normal distribution with mean $N{\pi }_t$ and variance obtained from the MAO norm, where ${\pi }_t=\left(\genfrac{}{}{0px}{}{T}{t}\right)p^t(1-p{)}^{T-t}$.2. Poisson approximation: When the expected value ${\lambda }_N:=N{\pi }_t\to 0$, $X_{=t}$ converges in distribution to $\mathrm{P}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{n}(\lambda )$. By symmetry, if ${\lambda }_N\to N$, then $N-X_{=t}$ converges to a Poisson distribution. All theoretical results are rigorously proved via moment methods and are corroborated by extensive numerical simulations, which further demonstrate that the approximations remain valid even when the subset sizes are not equal. The MAO norm thus emerges as a unifying tool that connects the exact combinatorial structure of the model to its asymptotic theory. These findings offer clear guidance for choosing the appropriate approximation in practical applications, and they open the door to using the MAO distribution as a null model for detecting non random aggregation in fields as diverse as ecology, sociology, and statistical physics.