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Convergence in a Multidimensional Randomized Keynesian Beauty Contest

  • Michael Grinfeld (a1), Stanislav Volkov (a2) and Andrew R. Wade (a3)
Abstract

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 ξ N ∈ [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 ξ N , 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.

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Corresponding author
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: andrew.wade@durham.ac.uk
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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
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