Probabilistic Stirling numbers and applications

We introduce probabilistic Stirling numbers of the first kind \(s_Y(n,k)\) associated with a complex-valued random variable Y satisfying appropriate integrability conditions, thus completing the notion of probabilistic Stirling numbers of the second kind \(S_Y(n,k)\) previously considered by the first author. Combinatorial interpretations, recursion formulas, and connections between \(s_Y(n,k)\) and \(S_Y(n,k)\) are given. We show that such numbers describe a large subset of potential polynomials, on the one hand, and the moments of sums of i. i. d. random variables, on the other, establishing their precise asymptotic behavior without appealing to the central limit theorem. We explicitly compute these numbers when Y has a certain familiar distribution, providing at the same time their combinatorial meaning.