Functional limit theorems for multitype branching processes and generalized Pَlya urns

A functional limit theorem is proved for multitype continuous time Markov branching processes. As consequences, we obtain limit theorems for the branching process stopped by some stopping rule, for example when the total number of particles reaches a given level. the Athreya–Karlin embedding, these results yield asymptotic results for generalized Pólya urns. We investigate such results in detail and obtain explicit formulas for the asymptotic variances and covariances. The general formulas involve integrals of matrix functions; we show how they can be evaluated and simplified in important special cases. We also consider the numbers of drawn balls of different types and functional limit theorems for the urns. ustrate our results by some examples, including several applications to random trees where our theorems and variance formulas give simple proofs of some known results; we also give some new results.