Central Limit Theorem for a Pólya-Friedman Mixed Urn Model

This paper considers a two-color, single-draw urn model with two types of balls, denoted type $1$ and type $2$, with initial counts $Y_0^1\in N^{+}$ and $Y_0^2\in N^{+}$, respectively. At each discrete time step, a ball is drawn uniformly at random, its type observed, and then it is returned to the urn. The urn is subsequently updated according to a mixed replacement matrix: with fixed probability $p\in (0,1)$, the Friedman replacement matrix is applied, adding $a$ balls of the drawn type and $b$ balls of the opposite type; with fixed probability $1-p\in (0,1)$, the Pólya replacement matrix is applied, adding $c$ balls of the drawn type. We establish the central limit theorem for the proportion of type $1$ balls after $n$ draws. Furthermore, we provide corollaries that yield large deviation inequalities and the law of the iterated logarithm related to the proportion of type $1$ balls after $n$ draws.