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

中文翻译：

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多色 Pólya 瓮模型的高斯过程近似

受数学组织生长建模的启发，我们考虑了近似多色 Pólya 瓮过程的动力学问题，该过程从大量不同颜色的球开始并长时间运行。使用经验和分位数过程的强逼近定理，我们为 Pólya 瓮过程建立高斯过程逼近。近似过程是多元布朗运动过程和具有随机高斯系数的独立线性漂移的总和。两者之间的主导项取决于时间步数的比率n到初始球数ñ在骨灰盒里。我们还建立了形式的上限$c(n^{-1/2}+N^{-1/2})$对于阶跃的凸 Borel 集类的最大偏差n瓮成分分布从近似正态法则。