Functional limit theorems for the Polya and q-Polya urns

For the plain Polya urn with two colors, black and white, we prove a functional central limit theorem for the number of white balls assuming that the initial number of black balls is large. Depending on the initial number of white balls, the limit is either a pure birth process or a diffusion. We also prove analogous results for the q-Polya urn, which is an urn where, when picking a ball, the balls of one color have priority over those of the other.