  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.
