A note on P\'olya urns: the winner may lead all the time
Svante Janson

TL;DR
This paper provides a simple proof that in a Pólya urn with certain initial conditions and replacement rules, the dominant color can maintain its lead indefinitely with positive probability.
Contribution
It introduces a straightforward proof demonstrating that the initial majority color can always stay ahead in a Pólya urn under specific conditions.
Findings
If the initial count of color 1 exceeds color 2, and m_1 ≥ m_2, then color 1 can always lead with positive probability.
The proof applies to two-color Pólya urns with fixed replacement numbers.
The result highlights the persistence of initial dominance in reinforced urn processes.
Abstract
Consider a P\'olya urn where a drawn ball of colour is replaced together with a fixed number of balls of the same colour. We give a simple proof that if, for example, there are two colours and the urn starts with more balls of colour 1 than 2, and , then there is a positive probability that there always will be more balls of colour 1 than colour 2.
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Taxonomy
TopicsLiterary, Cultural, Historical Analysis · Genome Rearrangement Algorithms · Limits and Structures in Graph Theory
