Limit laws for a class of diminishing urn models.

International audience In 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 $c=0$.