### Abstract

^{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.

Language | English |
---|---|

Pages | 57-82 |

Number of pages | 26 |

Journal | Advances in Applied Probability |

Volume | 47 |

Issue number | 1 |

DOIs | |

State | Published - 31 Mar 2015 |

### Fingerprint

### Keywords

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

### Cite this

*Advances in Applied Probability*,

*47*(1), 57-82. DOI: 10.1017/S0001867800007709

}

*Advances in Applied Probability*, vol. 47, no. 1, pp. 57-82. DOI: 10.1017/S0001867800007709

**Convergence in a multidimensional randomized Keynesian beauty contest.** / Grinfeld, Michael; Volkov, Stanislav; Wade, Andrew R.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Convergence in a multidimensional randomized Keynesian beauty contest

AU - Grinfeld,Michael

AU - Volkov,Stanislav

AU - Wade,Andrew R.

PY - 2015/3/31

Y1 - 2015/3/31

N2 - 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.

AB - 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.

KW - Keynesian beauty contest

KW - radius of gyration

KW - rank-driven process

KW - sum of squared distances

U2 - 10.1017/S0001867800007709

DO - 10.1017/S0001867800007709

M3 - Article

VL - 47

SP - 57

EP - 82

JO - Advances in Applied Probability

T2 - Advances in Applied Probability

JF - Advances in Applied Probability

SN - 0001-8678

IS - 1

ER -