Large deviations for Generalized Polya Urns with arbitrary urn function

Abstract

We consider a generalized two-color Polya urn (black and withe balls) first introduced by Hill, Lane, Sudderth where the urn composition evolves as follows: let $\pi:\left[0,1\right]\rightarrow\left[0,1\right]$, and denote by $x_{n}$ the fraction of black balls after step $n$, then at step $n+1$ a black ball is added with probability $\pi\left(x_{n}\right)$ and a white ball is added with probability $1-\pi\left(x_{n}\right)$. Originally introduced to mimic attachment under imperfect information, this model has found applications in many fields, ranging from Market Share modeling to polymer physics and biology. In this work we discuss large deviations for a wide class of continuous urn functions $\pi$. In particular, we prove that this process satisfies a Sample-Path Large Deviations principle, also providing a variational representation for the rate function. Then, we derive a variational representation for the limit $\phi\left(s\right)=\lim_{n\rightarrow\infty}{\textstyle \frac{1}{n}}\log\mathbb{P}\left(\left\{ nx_{n}=\left\lfloor sn\right\rfloor \right\} \right),\, s\in\left[0,1\right]$, where $nx_{n}$ is the number of black balls at time $n$, and use it to give some insight on the shape of $\phi\left(s\right)$. Under suitable assumptions on $\pi$ we are able to identify the optimal trajectory. We also find a non-linear Cauchy problem for the Cumulant Generating Function and provide an explicit analysis for some selected examples. In particular we discuss the linear case, which embeds the Bagchi-Pal Model, giving the exact implicit expression for $\phi$ in terms of the Cumulant Generating Function.