Explicit characterization of moments of balanced triangular Pólya urns by an elementary approach

Abstract

Two-color triangular urn models have been investigated recently. Moments of ball counts of a particular color have been obtained exactly and asymptotically in a number of sources (Janson, 2006, Kuba and Panholzer, 2014, and Flajolet et al., 2006). Exact factorial moments are in Kuba and Panholzer (2014). While Flajolet et al. (2006) gives an exact distribution, finding the exact moments from it is daunting. The asymptotic moments in Janson (2006) are derived by heavy machinery—the limit distribution in Janson (2006) is extracted from complicated characteristic functions in the form of integrals that they themselves include functions given by complicated integrals. The methods used in most of these investigations are quite sophisticated. In this note, we characterize the exact moments via an elementary approach. The formulas for the exact moments include Stirling’s numbers of second kind and the asymptotic moments follow effortlessly.