On martingale tail sums in affine two-color urn models with multiple drawings

TL;DR

This paper investigates the asymptotic behavior of martingale tail sums in affine two-color urn models with multiple drawings, establishing Gaussian limits, laws of the iterated logarithm, and properties of the martingale limits.

Contribution

It extends the understanding of tail sums in affine urn models, proving new limit laws and tail properties, especially for large-index and triangular urns.

Findings

Martingale tail sums follow Gaussian CLT in large-index urns.

Laws of the iterated logarithm are established for these tail sums.

Martingale limits have densities and exponentially decaying tails.

Abstract

In two recent works, Kuba and Mahmoud (arXiv:1503.090691 and arXiv:1509.09053) introduced the family of two-color affine balanced Polya urn schemes with multiple drawings. We show that, in large-index urns (urn index between $1/2$ and $1$) and triangular urns, the martingale tail sum for the number of balls of a given color admits both a Gaussian central limit theorem as well as a law of the iterated logarithm. The laws of the iterated logarithm are new even in the standard model when only one ball is drawn from the urn in each step (except for the classical Polya urn model). Finally, we prove that the martingale limits exhibit densities (bounded under suitable assumptions) and exponentially decaying tails. Applications are given in the context of node degrees in random linear recursive trees and random circuits.