Convergence in a multidimensional randomized Keynesian beauty contest

TL;DR

This paper analyzes a stochastic particle system where the farthest particle from the center is replaced by a random point, showing convergence to a configuration with most particles coinciding at a random location, with detailed results in one dimension.

Contribution

It introduces a new Markovian model of particle dynamics with convergence analysis and explicit distributional results in one dimension, extending understanding of multidimensional stochastic processes.

Findings

Particles converge to a configuration with N-1 coincident points.

The limiting point distribution in 1D assigns positive probability to any interval.

Explicit characterization of the limit for N=3 in one dimension.

Abstract

We study the asymptotics of a Markovian system of $N \geq 3$ 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 $\xi_N \in [0,1]^d$. 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 $\xi_N$, 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.