Large deviations for Generalized Polya Urns with non-binary increments

TL;DR

This work tackles the problem of extending the sample-path large deviation principle for Hill–Lane–Sudderth urns to increments that can take more than two values. It develops a path-integral, Lagrangian framework built on an urn-vector $\pi_k(\alpha)$ and the averaged increment $\bar{\pi}(\alpha)$, derives the scaling limit, and provides a variational characterization of the entropy density via Varadhan’s lemma and Mogulskii’s theorem. The authors derive the canonical scaled Lagrangian density $L(\alpha,\beta)$ and the baseline $L_0(\alpha)$, and give explicit results for the cases $K=1$ and $K=2$, including a gauge shift to align with free-energy conventions. These results enable synthetic data generation and benchmarking for urn-based stochastic systems and establish a general framework for non-binary, nonlinear Polya urn dynamics with potential applications to statistical physics and learning models. The approach broadens the applicability of SPLDP to multi-valued urn processes and provides concrete formulas and examples that can be used for theoretical analysis and computational experiments.

Abstract

In this paper we show how to extend the Sample-Path Large Deviation Principle for the urn model of Hill, Lane and Sudderth to the case in which the increment of the urn is not a binary variable. In particular, we sketch how to modify the Theorem 1 given in [Stochastic Processes and their Applications 127 (2017) 3372-3411] to include also urn processes with increments taking more than two values.