Probabilistic Stirling Numbers of the Second Kind and Applications
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Probabilistic Stirling Numbers of the Second Kind and Applications José A. Adell1 Received: 1 September 2020 / Revised: 15 October 2020 / Accepted: 17 October 2020 © The Author(s) 2020
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. Keywords Probabilistic Stirling number · Moment · Edgeworth expansion · Lévy process · Cumulant · Generalized difference Mathematics Subject Classification 60E05 · 05A19
1 Introduction The classical Stirling numbers play an important role in many branches of mathematics and physics as ingredients in the computation of diverse quantities. In particular, the Stirling numbers of the second kind S( j, m), counting the number of partitions of {1, . . . , j} into m nonempty, pairwise disjoint subsets, are a fundamental tool in many combinatorial problems. Such numbers can be defined in various equivalent ways (cf. Abramowitz and Stegun [1, p. 824] and Comtet [7, Chap. 5]). Two of the most useful are the following. Let j = 0, 1, . . . and m = 0, 1, . . . , j. Then S( j, m) can be explicitly defined as
B 1
José A. Adell [email protected] Departamento de Métodos Estadísticos. Facultad de Ciencias, Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain
123
Journal of Theoretical Probability m 1 m (−1)m−k k j , S( j, m) = k m!
(1)
k=0
or via their generating function as ∞ zj (e z − 1)m = S( j, m) , z ∈ C. m! j!
(2)
j=m
Motivated by various specific problems, different generalizations of the Stirling numbers S( j, m) have been considered in the literature (see, for instance, Hsu and Shiue [11], Luo and Srivastava [13], Caki´c et al. [6] and El-Desouky et al. [9]). In [3], we considered the following probabilistic generalization. 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). Then, the Stirling numbers of the second kind associated with Y are defined by SY ( j, m) =
m 1 m j (−1)m−k ESk , k m!
j = 0, 1, . . . , m = 0, 1, . . . , j. (3)
k=0
Observe that formula (3) recovers (1) when Y = 1. The motivations behind definition (3) have to do with certain problems coming from analytic number theory, such as extensions in various ways of the classical formula for sums of powers on arithmetic progressions (cf. [3]) and explicit expressions for higher-order convolutions of Appell polynomials (see [4]). In this paper, we extend definition (3) to complex-valued random variables Y and
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