Gaussian process approximations for multicolor Pólya urn models

Motivated by mathematical tissue growth modelling, we consider the problem of approximating dynamics multicolor P\'olya urn processes that start with large numbers balls different colors and run for a long time. Using strong approximation theorems empirical quantile processes, establish Gaussian process approximations processes. The are sums multivariate Brownian motion an independent linear drift random coefficient. Which two terms dominates depends on ratio number time steps $n$ to initial $N$ in urn. We also upper bound form $c(n^{-1/2}+N^{-1/2})$ maximum deviation over class convex Borel sets step composition distribution from normal law.