Functional limit theorems for the Pólya urn (Q2079163)

For the plain Pólya urn with two colors, black and white, the authors prove a functional central limit theorem for the number of white balls, assuming that the initial number of black balls is large. The aim is to examine whether the entire path \((A_n)\) of the urn scheme, after appropriate natural transformations, converges in distribution to a nontrivial stochastic process. The initial numbers of white and black balls depend on an additional parameter, and three regimes with respect to this parameter are considered. Namely, the number of white balls can be fixed, it can grow sublinearly and it can grow linearly. Depending on this behaviour of the initial number of white balls, the limit is either a pure birth process or a diffusion.