### 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.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 | |

Publication status | Published - 31 Mar 2015 |

### Keywords

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

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## Cite this

Grinfeld, M., Volkov, S., & Wade, A. R. (2015). Convergence in a multidimensional randomized Keynesian beauty contest.

*Advances in Applied Probability*,*47*(1), 57-82. https://doi.org/10.1017/S0001867800007709