On the long edges in the shortest tour throughN random points

Consider the shortest tour throughn pointsX ₁,...,X _n independently uniformly distributed over [0,1]². Then we show that for some universal constantK, the number of edges of length at leastun ^–1/2 is at mostKnxp(–u)²/K)with overwhelmingprobability.This research is in part supported by an NSF grant.     

Uniform bound in the central limit theorem for Banach space valued dependent random variables

Let F be a Banach space with a sufficiently smooth norm. Let (X_i)_i≤n be a sequence in L_F², and T be a Gaussian random variable T which has the same covariance as X = Σ_i≤nX_i. Assume that there exists a constant G such that for s, δ≥0, we have P(sTs+δ)Gδ. (*) We then give explicit bounds of Δ(X) = sup_i|P(|X|≤t)−P(|T|≤t)| in terms of truncated moments of the variables X_i. These bounds hold under rather mild weak dependence conditions of the variables. We also construct a Gaussian random variable that violates (^*).