On the area under lattice paths associated with triangular diminishing urn models

Abstract

This work is devoted to the analysis of the area under certain lattice paths. The lattice paths of interest are associated to a class of 2 × 2 triangular Pólya–Eggenberger urn models with ball replacement matrix M = ( − a 0 c − d ) , with a , d ∈ N and c = p ⋅ a , p ∈ N 0 . We study the random variable counting the area under sample paths associated to these urn models, where we obtain a precise recursive description of its positive integer moments. This description allows us to derive exact formulae for the expectation and the variance and, in principle, also for higher moments and, most nobably, it yields asymptotic expansions of all positive integer moments leading to a complete characterization of the limiting distributions appearing for the area under sample paths associated with these urn models. As a special instance we obtain limiting distributions for the area under sample paths of the pills problem urn model, originally proposed by Knuth and McCarthy, which corresponds to the special case a = c = d = 1 . Furthermore we also obtain limiting distributions for the well-known sampling without replacement urn, a = d = 1 and c = 0 , and generalizations of it to a , d ∈ N .