There has been much interest recently in the specially constructed empirical processes of Komlós, Major and Tusnády [2]; as one would guess, much of the application has come from the Hungarian school. In this note we contribute to the unifying effect this profound work has had by showing how...

NSF, NIAID

Distributions of functionals of Brownian bridge arise as limiting distributions in non-parametric statistics. In this paper we will give a derivation of distributions of extrema of the Brownian bridge based on excursion theory for Brownian motion. The idea of rescaling and conditioning on the...

We establish upper and lower bounds for the metric entropy and bracketing entropy of the class of d-dimensional bounded monotonic functions under Lp norms. It is interesting to see that both the metric entropy and bracketing entropy have different behaviors for p<d/(d-1) and p>d/(d-1). We apply the new bounds...</d/(d-1)>

For n particles diffusing throughout R (or Rd), let [eta]n,t(A), A [epsilon] B, t [greater-or-equal, slanted]0, be the random measure that counts the number of particles in A at time t. It is shown that for some basic models (Brownian particles with or without branching and diffusion with a...

The bounded-dual-Lipschitz and Prohorov distances from the 'empirical measure' to the 'average measure' of independent random variables converges to zero almost surely if the sequence of average measures is tight. Three examples are also given.

Suppose that U=(U1,…,Ud) has a Uniform([0,1]d) distribution, that Y=(Y1,…,Yd) has the distribution G on R+d, and let X=(X1,…,Xd)=(U1Y1,…,UdYd). The resulting class of distributions of X (as G varies over all distributions on R+d) is called the Scale Mixture of Uniforms class of...

Suppose one observes a sample of size m from the mixture density [integral operator] p(xz) d[eta](z) and a sample of size n from the distribution [eta]. The kernel p(xz) is known. We show existence of the maximum likelihood estimator for [eta], characterize its support, and prove consistency as...