Convergence in a multidimensional randomized Keynesian beauty contest

We study the asymptotics of a Markovian system of $N\ge 3$ 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 $U [0,1]^d$randomparticle\mathrm{.}Weshowthatthelimitingconfigurationcontains$N-1$coincidentparticlesatarandomlocation$\xi_N \in [0,1]^d$. 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.