441 research outputs found
Worst case asymptotics of power-weighted Euclidean functionals
AbstractLet A,|A|⩽n, be a subset of [0,1]d, and let L(A,[0,1]d,p) be the length of the minimal matching, the minimal spanning tree, or the traveling salesman problem on A with weight function w(e)=|e|p. In the case 1⩽p<d, Yukich (Combinatorica 16 (1996) 575) obtained the asymptotic of αL(n,d,p)=maxA⊂[0,1]d,|A|⩽nL(A,[0,1]d,p). In this paper we extend his result to the whole range 0<p<∞
Stability and integration over Bergman metrics
We study partition functions of random Bergman metrics, with the actions
defined by a class of geometric functionals known as `stability functions'. We
introduce a new stability invariant - the critical value of the coupling
constant - defined as the minimal coupling constant for which the partition
function converges. It measures the minimal degree of stability of geodesic
rays in the space the Bergman metrics, with respect to the action. We calculate
this critical value when the action is the -balancing energy, and show
that on a Riemann surface of genus .Comment: 24 pages, 3 figure
A Riemannian-Stein Kernel Method
This paper presents a theoretical analysis of numerical integration based on
interpolation with a Stein kernel. In particular, the case of integrals with
respect to a posterior distribution supported on a general Riemannian manifold
is considered and the asymptotic convergence of the estimator in this context
is established. Our results are considerably stronger than those previously
reported, in that the optimal rate of convergence is established under a basic
Sobolev-type assumption on the integrand. The theoretical results are
empirically verified on
Connected Spatial Networks over Random Points and a Route-Length Statistic
We review mathematically tractable models for connected networks on random
points in the plane, emphasizing the class of proximity graphs which deserves
to be better known to applied probabilists and statisticians. We introduce and
motivate a particular statistic measuring shortness of routes in a network.
We illustrate, via Monte Carlo in part, the trade-off between normalized
network length and in a one-parameter family of proximity graphs. How close
this family comes to the optimal trade-off over all possible networks remains
an intriguing open question. The paper is a write-up of a talk developed by the
first author during 2007--2009.Comment: Published in at http://dx.doi.org/10.1214/10-STS335 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
- …