Given a family of k + 1 real-valued functions f0,...,f k defined on the set {1,...,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 T0,...,Tk, the size a of the collection of numbers j ∈ {1,...,n} whose signals f0(j),...,fk(j) exceed the corresponding threshold values T0,...,Tk simultaneously for all 0,...,k is surprisingly large (or small) in comparison to the family of cardinalities ai := #{j ∈ {1,...,n}|fi(j) > T i} (i = 0,...,k) of those numbers 1 ≤ j ≤ n whose signals fi(j) individually exceed, for a given index i, the corresponding threshold value Ti. 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|a0,...,ak (a) of families of subsets A0, A 1,...,Ak of {1,...,n} with #Ai = ai for all i = 0,...,k and #∩i=0,...,kAi = a, and to investigate their asymptotic behaviour. In this note, we show that the associated probability distributions pn|a0,...,ak = (p n|a0,...,ak (a))a∈N0 defined on the set N0 of non-negative integers by (Equation Presented) converge, with n → ∞, towards the Poisson distribution poissα = (poiss α(a))a∈N0 for some fixed α ∈ R >0 provided the numbers ai (i = 0,...,k) are assumed to converge with n to infinity in such a way that the conditions (Equation Presented) are satisfied. Remarkably, it is the alternating signs in the expressions for An|a0,...,ak (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.
