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A restaurant process with cocktail bar and relations to the three-parameter Mittag–Leffler distribution

Part of: Sequences and sets Combinatorial probability Markov processes Limit theorems Other special functions

Published online by Cambridge University Press:  22 November 2021

Martin Möhle
Martin Möhle*
Affiliation:
Eberhard Karls Universität Tübingen
*
*Postal address: Mathematisches Institut, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany. Email address: martin.moehle@uni-tuebingen.de
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Abstract

In addition to the features of the two-parameter Chinese restaurant process (CRP), the restaurant under consideration has a cocktail bar and hence allows for a wider range of (bar and table) occupancy mechanisms. The model depends on three real parameters, $\alpha$ , $\theta_1$ , and $\theta_2$ , fulfilling certain conditions. Results known for the two-parameter CRP are carried over to this model. We study the number of customers at the cocktail bar, the number of customers at each table, and the number of occupied tables after n customers have entered the restaurant. For $\alpha>0$ the number of occupied tables, properly scaled, is asymptotically three-parameter Mittag–Leffler distributed as n tends to infinity. We provide representations for the two- and three-parameter Mittag–Leffler distribution leading to efficient random number generators for these distributions. The proofs draw heavily from methods known for exchangeable random partitions, martingale methods known for generalized Pólya urns, and results known for the two-parameter CRP.

Keywords

Beta distributionexchangeabilityoccupied tablesPólya urnrandom number generationrandom partitionrejection methodrestaurant processStirling numbersthree-parameter Mittag–Leffler distribution

MSC classification

Primary: 60C05: Combinatorial probability 60F05: Central limit and other weak theorems
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 33E12: Mittag-Leffler functions and generalizations 11B73: Bell and Stirling numbers
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 4 , December 2021 , pp. 978 - 1006
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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