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Random additions in urns of integers

Part of: Special processes Limit theorems

Published online by Cambridge University Press:  23 June 2021

Mackenzie Simper Open the ORCID record for Mackenzie Simper [Opens in a new window]
Mackenzie Simper*
Affiliation:
Stanford University
*
*Postal address: Department of Mathematics, 450 Jane Stanford Way, Building 380, Stanford, CA 94305-2125. Email address: [email protected].
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Abstract

Consider an urn containing balls labeled with integer values. Define a discrete-time random process by drawing two balls, one at a time and with replacement, and noting the labels. Add a new ball labeled with the sum of the two drawn labels. This model was introduced by Siegmund and Yakir (2005) Ann. Prob.33, 2036 for labels taking values in a finite group, in which case the distribution defined by the urn converges to the uniform distribution on the group. For the urn of integers, the main result of this paper is an exponential limit law. The mean of the exponential is a random variable with distribution depending on the starting configuration. This is a novel urn model which combines multi-drawing and an infinite type of balls. The proof of convergence uses the contraction method for recursive distributional equations.

Keywords

urn modelexponential limit lawrecursive distributional equationcontraction method

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 2 , June 2021 , pp. 335 - 346
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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