Probabilistic Stirling numbers of the second kind and applications

TL;DR

This paper introduces a new probabilistic generalization of Stirling numbers of the second kind linked to complex random variables, enabling precise asymptotic analysis and Edgeworth expansions for sums of i.i.d. variables, with applications to Lévy processes.

Contribution

It presents a novel probabilistic framework for Stirling numbers of the second kind, connecting them to moments of sums of i.i.d. random variables and their asymptotic behavior.

Findings

Provides explicit formulas for the generalized Stirling numbers.

Derives asymptotic behavior of sums of i.i.d. variables without CLT.

Develops simple Edgeworth expansions for these sums.

Abstract

Associated to each complex-valued random variable satisfying appropriate integrability conditions, we introduce a different generalization of the Stirling numbers of the second kind. Various equivalent definitions are provided. Attention, however, is focused on applications. Indeed, such numbers describe the moments of sums of i.i.d. random variables, determining their precise asymptotic behavior without making use of the central limit theorem. Such numbers also allow us to obtain explicit and simple Edgeworth expansions. Applications to L\'{e}vy processes and cumulants are discussed, as well.