Random additions in urns of integers

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...

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Published in:Journal of applied probability 2021-06, Vol.58 (2), p.335-346
Main Author: Simper, Mackenzie
Format: Article
Language:English
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Convergence
Expected values
Group theory
Integers
Labels
Original Article
Random processes
Random variables
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description 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.
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subjects Convergence
Expected values
Group theory
Integers
Labels
Original Article
Random processes
Random variables
title Random additions in urns of integers
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