Let $(\Omega,\mathcal{F},P)$ be a \reftext{def:probability-space-random-variable-2026a}{probability space}, $\mathbb{N}$ the set of \reftext{def:natural-numbers-2026a}{natural numbers} with $\mathbb{N}_0=\mathbb{N}\cup\{0\}$, and $\mathbb{R}$ the set of \reftext{def:real-numbers-c54-2026c}{real numbers}. Let $\mu\ge0$ be real, let $K$ be a random variable with the \reftext{def:poisson-distribution-2026b}{Poisson distribution} with parameter $\mu$, let $\nu$ be a probability \reftext{def:measure-measure-space-2026a}{measure} on the \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel $\sigma$-algebra}, and let $(V_i)_{i\in\mathbb{N}}$ be random variables each with \reftext{def:distribution-cdf-random-variable-2026a}{distribution} $\nu$, such that the \reftext{def:family-subfamily-subsets-set-2026a}{family} $(K,V_1,V_2,\dots)$ is \reftext{def:independence-events-rvs-2026a}{independent}.

Let $r\in\mathbb{N}$ and let $A_1,\dots,A_r$ be pairwise disjoint Borel sets, with $p_j=\nu(A_j)$. Define $\widetilde K(\omega)=K(\omega)$ if $K(\omega)\in\mathbb{N}_0$ and $\widetilde K(\omega)=0$ otherwise (so $\widetilde K=K$ with probability $1$, since a Poisson variable lies in $\mathbb{N}_0$ with probability $1$ by \ref{def:poisson-distribution-2026b}), and define the \textbf{cell counts}

where $\mathbf{1}_{E}$ is the function equal to $1$ on $E$ and $0$ off $E$, and an empty sum is $0$. Then:

\textbf{1.} each $C_j$ is a random variable with all values in $\mathbb{N}_0$, measurable with respect to the \reftext{def:independence-sigma-algebras-2026a}{generated $\sigma$-algebra} $\sigma\bigl(K,(V_i)_{i\in\mathbb{N}}\bigr)$;

\textbf{2.} for all $(n_1,\dots,n_r)\in\mathbb{N}_0^{r}$, with the \reftext{def:factorial-natural-number-2026a}{factorial} (convention $0!=1$), the convention $x^{0}=1$, the \reftext{def:exponential-function-real-2026a}{exponential function} $\exp$, and the \reftext{def:finite-product-notation-2026a}{finite product notation},

\textbf{3.} the random variables $C_1,\dots,C_r$ are independent, and $C_j$ has the Poisson distribution with parameter $\mu p_j$.