Two-color balanced affine urn models with multiple drawings II: large-index and triangular urns

Abstract

This is the second part of a two-part investigation. We continue the study of a class of balanced urn schemes on balls of two colors (white and black). At each drawing, a sample of size $m\ge 1$ is drawn from the urn and ball addition rules are applied; the special case $m=1$ of sampling only a single ball coincides with ordinary balanced urn models. We consider these multiple drawings under sampling with or without replacement. For the class of affine conditional expected value, we study the number of white balls after $n$ steps. The affine class is parametrized by $\Lambda$, specified by the ratio of the two eigenvalues of a reduced ball replacement matrix and the sample size, leading to three different cases: small-index urns ($\Lambda \le \frac 1 2$, and the case $\Lambda= \frac 1 2$ is critical), large-index urns ($\Lambda > \frac 1 2$), and triangular urns. In Part I we derived a central limit theorem for small index urns, and proved almost-sure convergence for large index and triangular urn models. In the present paper (Part II), we continue the study of affiance urn schemes and study the moments of large-index urns and triangular urn models. We show moment convergence under suitable scaling and we also provide expressions for the moments.