Poisson Numbers and Poisson Distributions in Subset Surprisology

Abstract.

Given a family of k + 1 real-valued functions $f_0 , \ldots ,f_k$ defined on the set $\{ 1, \ldots ,n\}$ and measuring the intensity of certain signals, we want to investigate whether these functions are ‘dependent’ or ‘independent’ by checking whether, for some given family of threshold values $T_0 , \ldots ,T_k ,$ the size a of the collection of numbers $j \in \{ 1, \ldots ,n\}$ whose signals $f_0 (j), \ldots ,f_k (j)$ exceed the corresponding threshold values $T_0 , \ldots ,T_k$ simultaneously for all $0, \ldots ,k$ is surprisingly large (or small) in comparison to the family of cardinalities

$$ a_i : = \# \{ j \in \{ 1, \ldots ,n\} |f_i (j) > T_i \} \;(i = 0, \ldots ,k) $$

of those numbers $1 \leq j \leq n$ whose signals f _i (j) individually exceed, for a given index i, the corresponding threshold value T _i . Such problems turn presently up in topological proteomics, a new direction of protein-interaction research that has become feasible due to new techniques developed in fluorescence microscopy called Multi-Epitope Ligand Cartography (or, for short, MELK = Multi-Epitop Liganden Kartographie). The above problem has led us to study the numbers $A_{n|a_0 , \ldots ,a_k } (a)$ of families of subsets $A_0 ,A_1 , \ldots ,A_k$ of $\{ 1, \ldots ,n\}$ with $\# A_i = a_i$ for all $i = 0, \ldots ,k$ and $\# \bigcap\nolimits_{i = 0, \ldots k} {A_i = a,}$ and to investigate their asymptotic behaviour. In this note, we show that the associated probability distributions

$$ p_{n|a_0 , \ldots ,a_k } = \left( {p_{n|a_0 , \ldots ,a_k } (a)} \right)_{a \in {\text{N}}_0 } $$

defined on the set ${\mathbf{N}}_0$ of non-negative integers by

$$ p_{n|a_0 , \ldots ,a_k } (a): = \frac{{A_{n|a_0 , \ldots ,a_k } (a)}} {{\prod\nolimits_{i = 0}^k {\left( {\begin{array}{*{20}c} n \\ {a_i } \\ \end{array} } \right)} }}\quad (a \in {\text{N}}_0 ) $$

converge, with n → ∞, towards the Poisson distribution

$$ {\text{poiss}}_\alpha = ({\text{poiss}}_\alpha (a))_{a \in {\mathbf{N}}_0 } $$

for some fixed $\alpha \in {\mathbf{R}}_{ > 0}$ provided the numbers $a_i \,(i = 0, \ldots ,k)$ are assumed to converge with n to infinity in such a way that the conditions

$$ a_i \leq n\quad {\text{and}}\quad \mathop {\lim }\limits_{n \to \infty } \frac{{\prod\nolimits_{i = 0}^k {a_i } }} {{n^k }} = \alpha $$

are satisfied. Remarkably, it is the alternating signs in the expressions for $A_{n|a_0 , \ldots ,a_k } (a)$ resulting from the standard exclusion-inclusion principle that correspond to the alternating signs in the power series expression for exp(−α) when n turns to infinity.