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Covariances in Pólya urn schemes

Part of: Limit theorems Combinatorial probability Special processes

Published online by Cambridge University Press:  08 October 2021

Hosam Mahmoud Open the ORCID record for Hosam Mahmoud [Opens in a new window]
Hosam Mahmoud*
Affiliation:
Department of Statistics, The George Washington University, Washington, D.C. 20052, USA. E-mail: hosam@gwu.edu
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Abstract

By now there is a solid theory for Polya urns. Finding the covariances is somewhat laborious. While these papers are “structural,” our purpose here is “computational.” We propose a practicable method for building the asymptotic covariance matrix in tenable balanced urn schemes, whereupon the asymptotic covariance matrix is obtained by solving a linear system of equations. We demonstrate the use of the method in growing tenable balanced irreducible schemes with a small index and in critical urns. In the critical case, the solution to the linear system of equations is explicit in terms of an eigenvector of the scheme.

Keywords

L2-limit theoremsPólya urn

MSC classification

Primary: 60C05: Combinatorial probability 60F25: $L^p$-limit theorems
Secondary: 60K40: Other physical applications of random processes
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 37 , Issue 1 , January 2023 , pp. 60 - 71
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

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Footnotes

To Robert Smythe, a mentor, coauthor and friend, on his 80th birthday

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