# Convergence in a multidimensional randomized Keynesian beauty contest

Research output: Research - peer-review › Article

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.Original language | English |
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Pages (from-to) | 57-82 |

Number of pages | 26 |

Journal | Advances in Applied Probability |

Volume | 47 |

Issue number | 1 |

DOIs | |

State | Published - 31 Mar 2015 |

- Keynesian beauty contest, radius of gyration, rank-driven process, sum of squared distances