PC Seminar: Strong limit laws for triangular Pólya urns

Abstract:

Consider a triangular Pólya urn, and let $X_{ni}$ be the number of balls of
colour $i$ after $n$ drawings. We show, under very mild technical conditions, that there exist constants $\beta_i \ge 0$ and $\gamma_i\in(-\infty,\infty)$ such that
$ X_{in}/(n^{\beta_i} (\log n)^{\gamma_i}) $ converges a.s. to some strictly positive random variable. The numbers $\beta_i$ and $\gamma_i$ have explicit definitions.

The limit random variable is sometimes degenerate (= a constant); we more or
less describe when this happens.

The proof is based on the old method of embedding in a continuous-time version
of the urn, and proving corresponding results in continuous time.

This is a seminar in our seminar series on Probability and Combinatorics (PC).