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

Part of: Stochastic processes Limit theorems Combinatorial probability

Published online by Cambridge University Press:  04 April 2017

Markus Kuba  and
Henning Sulzbach
Markus Kuba*
Affiliation:
University of Applied Sciences Technikum Wien
Henning Sulzbach*
Affiliation:
McGill University
*
* Postal address: Institute of Applied Mathematics and Natural Sciences, University of Applied Sciences-Technikum Wien, Höchstädtplatz 5, 1200 Wien, Austria. Email address: kuba@technikum-wien.at
** Current address: School of Mathematics, University of Birmingham, BirminghamB15 2TT, UK.
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Abstract

In two recent works, Kuba and Mahmoud (2015a) and (2015b) introduced the family of two-color affine balanced Pólya urn schemes with multiple drawings. We show that, in large-index urns (urn index between ½ 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 Pólya 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.

Keywords

Urn modelmartingale central limit theoremlaw of the iterated logarithmlarge-index urntriangular urn

MSC classification

Primary: 60F15: Strong theorems
Secondary: 60C05: Combinatorial probability 60F05: Central limit and other weak theorems 60G42: Martingales with discrete parameter
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 1 , March 2017 , pp. 96 - 117
Copyright
Copyright © Applied Probability Trust 2017 

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