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On affine multicolor urns grown under multiple drawing

Part of: Stochastic processes Limit theorems Combinatorial probability

Published online by Cambridge University Press:  24 December 2024

Joshua Sparks ,
Markus Kuba ,
Srinivasan Balaji  and
Hosam Mahmoud Open the ORCID record for Hosam Mahmoud [Opens in a new window]
Joshua Sparks*
Affiliation:
The George Washington University
Markus Kuba*
Affiliation:
FH Technikum Wien
Srinivasan Balaji*
Affiliation:
The George Washington University
Hosam Mahmoud*
Affiliation:
The George Washington University
*
* Postal address: Department of Statistics, Washington, D.C. 20052, USA
*** Postal address: Department of Applied Mathematics and Physics, Vienna, Austria. Email: [email protected]
* Postal address: Department of Statistics, Washington, D.C. 20052, USA
* Postal address: Department of Statistics, Washington, D.C. 20052, USA
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Abstract

Early investigation of Pólya urns considered drawing balls one at a time. In the last two decades, several authors have considered multiple drawing in each step, but mostly for schemes involving two colors. In this manuscript, we consider multiple drawing from urns of balls of multiple colors, formulating asymptotic theory for specific urn classes and addressing more applications. The class we consider is affine and tenable, built around a ‘core’ square matrix. We examine cases where the urn is irreducible and demonstrate its relationship to matrix irreducibility for its core matrix, with examples provided. An index for the drawing schema is derived from the eigenvalues of the core. We identify three regimes: small, critical, and large index. In the small-index regime, we find an asymptotic Gaussian law. In the critical-index regime, we also find an asymptotic Gaussian law, albeit with a difference in the scale factor, which involves logarithmic terms. In both of these regimes, we have explicit forms for the structure of the mean and the covariance matrix of the composition vector (both exact and asymptotic). In all three regimes we have strong laws.

Keywords

Pólya urncombinatorial probabilitylimiting distributionmultivariate central limit theoremmultivariate martingale

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60C05: Combinatorial probability 60G46: Martingales and classical analysis
Type
Original Article
Information
Journal of Applied Probability , Volume 62 , Issue 2 , June 2025 , pp. 808 - 831
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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