https://hal.inria.fr/hal-01184767Kuba, MarkusMarkusKubaInstitut für Diskrete Mathematik und Geometrie [Wien] - TU Wien - Vienna University of TechnologyPanholzer, AloisAloisPanholzerInstitut für Diskrete Mathematik und Geometrie [Wien] - TU Wien - Vienna University of TechnologyLimit laws for a class of diminishing urn models.HAL CCSD2007Pólya urnsdiminishing urnspills problemsampling without replacement[INFO.INFO-DS] Computer Science [cs]/Data Structures and Algorithms [cs.DS][INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM][MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO][INFO.INFO-CG] Computer Science [cs]/Computational Geometry [cs.CG]Episciences Iam, CoordinationJacquet, Philippe2015-08-17 16:58:422021-10-13 19:58:042015-08-24 10:04:14enConference papershttps://hal.inria.fr/hal-01184767/document10.46298/dmtcs.3519application/pdf1In this work we analyze a class of diminishing 2×2 Pólya-Eggenberger urn models with ball replacement matrix M given by \$M= \binom{ -a \,0}{c -d}, a,d∈\mathbb{N}\$ and \$c∈\mathbb{N} _0\$. We obtain limit laws for this class of 2×2 urns by giving estimates for the moments of the considered random variables. As a special instance we obtain limit laws for the pills problem, proposed by Knuth and McCarthy, which corresponds to the special case \$a=c=d=1\$. Furthermore, we also obtain limit laws for the well known sampling without replacement urn, \$a=d=1\$ and \$c=0\$, and corresponding generalizations, \$a,d∈\mathbb{N}\$ and \$c=0\$.