Empirical process of continuous-time Markov branching processes and applications to Polya's urn

Vidyashankar Anand, George Mason University

Polya's urn model and its variants have received much attention recently and are used to model various social and biological phenomena. Continuous time multitype Markov branching processes (CTMMBP) are frequently used to prove limit theorems regarding urn models, preferential attachment trees, and random recursive trees. Additionally, these processes are used to identify the proportion of distinct types in cell kinetics. In this presentation, we describe the empirical process of generation sizes of CTMMBP and related uniform laws of large numbers and uniform central limit theorem under random metric entropy conditions. Using these results and the Athreya-Karlin embedding technique, we derive related limit theorems for Polya's urn process and preferential attachment trees. We also present some applications to response adaptive randomization in clinical trials.

Area: CS40 - Recent developments for urn models (Andrea Ghiglietti and Giacomo Aletti)

Keywords: Markov branching process, metric entropy, Polya's urn

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