Functional limit theorems for P\'olya urns with growing initial compositions

Abstract

In this paper, we prove functional limit theorems for P\'olya urn processes whose number of draws and initial number of balls tend to infinity together. This is motivated by recent work of Borovkov [5], where they prove a functional limit theorem for this model when the urn has identity replacement rule. We generalize this result to arbitrary balanced replacement rules (the total number of balls added to the urn is deterministic). Three asymptotic regimes are possible depending on how one lets the number of initial balls scale with the number of draws of the urn. In each regime, we show a first order deterministic limit and Gaussian second order fluctuations, where the behaviour of these limit processes depend on the regime, the initial composition of the urn, and the urns replacement rule. To prove our main results, we embed the process in continuous-time and use martingale theory. Although these methods are classical since the works of Athreya & Karlin [1] and Janson [11], our setting with initial growing composition necessitates many new ideas. The main difference in proving limiting results for the continuous time embedding in our setting is that, when the initial composition is large compared to the number of draws, the branching process does not have time to reach equilibrium. Because of this, translating the results back to discrete-time is also much harder than in Janson [11]. Interestingly, our continuous-time results hold under weaker assumptions on the replacement structure than classical results for multi-type branching processes; in particular, we do not need any "irreducibility" assumption.