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Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Harvard Univ, Dept Stat, 1 Oxford St,SC 712, Cambridge, MA 02138 USA.
NYU, Courant Inst, Dept Comp Sci, Room 829,251 Mercer St, New York, NY 10012 USA;NYU, Courant Inst, Dept Math, Room 829,251 Mercer St, New York, NY 10012 USA.
2019 (English)In: Advances in Applied Probability, ISSN 0001-8678, E-ISSN 1475-6064, Vol. 51, no 4, p. 1067-1108Article in journal (Refereed) Published
##### Abstract [en]

We study an urn process with two urns, initialized with a ball each. Balls are added sequentially, the urn being chosen independently with probability proportional to the ath power (alpha > 1) of the existing number of balls. We study the (rare) event that the urn compositions are balanced after the addition of 2n - 2 new balls. We derive precise asymptotics of the probability of this event by embedding the process in continuous time. Quite surprisingly, fine control of this probability may be leveraged to derive a lower- tail large deviation principle (LDP) for L= Sigma(n)(i=1) (S-i(2)/i(2)), where {S-n : n >= 0} is a simple symmetric random walk started at zero. We provide an alternative proof of the LDP via coupling to Brownian motion, and subsequent derivation of the LDP for a continuous-time analog of L. Finally, we turn our attention back to the urn process conditioned to be balanced, and provide a functional limit law describing the trajectory of the urn process.

##### Place, publisher, year, edition, pages
APPLIED PROBABILITY TRUST , 2019. Vol. 51, no 4, p. 1067-1108
##### Keywords [en]
Urn model, large deviations
##### National Category
Probability Theory and Statistics
##### Identifiers
ISI: 000496968500005OAI: oai:DiVA.org:uu-398564DiVA, id: diva2:1376365
##### Funder
Knut and Alice Wallenberg FoundationAvailable from: 2019-12-09 Created: 2019-12-09 Last updated: 2019-12-09Bibliographically approved

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Janson, Svante

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Probability Theory and Statistics

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Cite
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