Convergence rates of minimal graphs with random vertices", (submitted to

This paper is concerned with power-weighted weight functionals associated with a minimal graph span-ning a random sample of n points from a general multivariate Lebesgue density f over [0; 1]d. It is known that under broad conditions, when the functional applies power exponent 2 (1; d) to the graph edge lengths, the log of the functional normalized by n(d )=d is a strongly consistent estimator of the Rényi entropy of order = (d )=d. In this paper we investigate almost sure (a.s.) and L-norm (r.m.s. for = 2) convergence rates of this functional. In particular, when 1 d 1, we show that over the space of compacted supported multivariate densities f such that f 2W 1;p(Rd) (the space of Sobolev func-tions) the L-norm convergence rate is bounded above by