Abstract
Associated with 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 Lévy processes and cumulants are discussed, as well.
Highlights
The classical Stirling numbers play an important role in many branches of mathematics and physics as ingredients in the computation of diverse quantities
Let (Yk)k≥1 be a sequence of independent copies of a real-valued random variable Y having a finite moment generating function and denote by Sk = Y1 + · · · + Yk, k = 1, 2, . . . (S0 = 0)
Let G0 be the set of complex-valued random variables Y having a finite moment generating function in a neighborhood of the origin, i.e., Ee|zY | < ∞, |z| < R, for some R > 0
Summary
The classical Stirling numbers play an important role in many branches of mathematics and physics as ingredients in the computation of diverse quantities. Let (Yk)k≥1 be a sequence of independent copies of a real-valued random variable Y having a finite moment generating function and denote by Sk = Y1 + · · · + Yk, k = 1, 2, . We extend definition (3) to complex-valued random variables Y and show its usefulness in various classical topics of probability theory. 5, we give explicit and relatively simple full Edgeworth expansions whose coefficients depend on the Stirling numbers SY +i Z ( j, m), where the real-valued random variables.
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