Long Run Behaviour of Infinite Pólya Urn Models

Systems that preserve and amplify existing inequalities are said to be characterized by richget-richer dynamics, where advantages obtained early on are reinforced. Pólya urn models
are a key element of studying such random processes with reinforcement.
The basic urn model starts with an urn containing balls of finitely many colours which
evolves in discrete time. At each step, we shake the urn well and a ball is sampled
uniformly at random and returned to the urn. Then according to the colour of this ball, we
use a schema (represented as a matrix) to decide how many balls of what colour to be
placed in the urn.
While the basic Pólya urn has a random limit depending critically on the initial conditions,
generalised Pólya urn models can have deterministic and stable asymptotic behaviours in
the long run. Such a generalised urn model can be studied by embedding it into a
continuous time Markov branching process, and it is possible for us to understand the longrun behaviour using Law of Large Numbers, Central Limit Theorem and other approaches.
Recently, a new type of model for balanced urn schemes with countably or incountably
infinitely many colours has been introduced. The goal of the project is to study these
infinite P ́olya urn models and explore their large time behaviour in analogy to the finite
case.

Jiani Xie

The University of Melbourne

Jiani Xie is a student at the University of Melbourne having just completed her final-year study for a Bachelor of Science (Computer Science major) with a Concurrent Diploma in Mathematical Sciences (Statistics/Stochastic Processes path). As an interdisciplinary learner, she enjoys absorbing the knowledge in both fields and always finds it fascinating when they interact and connect with each other. Last year, Jiani had an opportunity to work on a supervised summer project in applied statistics, which evaluated and analysed the performance of deep learning models. The experience sparked her enthusiasm for mathematical research and inspired her to pursue further studies in probability models together with various areas related. Jiani is interested in stochastic modelling and random processes in particular and would like to learn more about pure mathematics as well. Outside of her studies, Jiani loves music, travelling, reading, and taking a walk-in nature. Being a math’s tutor for more than two years, she has developed a strong interest in teaching, eager to help people who want to know more about mathematics in the future.

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