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Convergence in a Multidimensional Randomized Keynesian Beauty Contest

Part of: Special processes Limit theorems Game theory Time-dependent statistical mechanics (dynamic and nonequilibrium) Markov processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  04 January 2016

Michael Grinfeld ,
Stanislav Volkov  and
Andrew R. Wade
Michael Grinfeld*
Affiliation:
University of Strathclyde
Stanislav Volkov*
Affiliation:
Lund University and University of Bristol
Andrew R. Wade*
Affiliation:
Durham University
*
∗ Postal address: Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK.
∗∗ Postal address: Centre for Mathematical Sciences, Lund University, Box 118, Lund, SE-22100, Sweden.
∗∗∗ Postal address: Department of Mathematical Sciences, Durham University, South Road, Durham DH1 3LE, UK. Email address: [email protected]
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Abstract

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We study the asymptotics of a Markovian system of N ≥ 3 particles in [0, 1]d in which, at each step in discrete time, the particle farthest from the current centre of mass is removed and replaced by an independent U[0, 1]d random particle. We show that the limiting configuration contains N − 1 coincident particles at a random location ξN ∈ [0, 1]d. A key tool in the analysis is a Lyapunov function based on the squared radius of gyration (sum of squared distances) of the points. For d = 1, we give additional results on the distribution of the limit ξN, showing, among other things, that it gives positive probability to any nonempty interval subset of [0, 1], and giving a reasonably explicit description in the smallest nontrivial case, N = 3.

Keywords

Keynesian beauty contestradius of gyrationrank-driven processsum of squared distances

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60D05: Geometric probability and stochastic geometry 60F15: Strong theorems 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82C22: Interacting particle systems 91A15: Stochastic games
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 1 , March 2015 , pp. 57 - 82
Copyright
© Applied Probability Trust 

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