Georgia Institute of Technology College of Sciences

We consider random walks on Z^d among nearest-neighbor random conductances which are i.i.d., positive, bounded uniformly from above but which can be arbitrarily close to zero. Our focus is on the detailed properties of the paths of the random walk conditioned to return back to the starting point after time 2n. We show that in the situations when the heat kernel exhibits subdiffusive behavior --- which is known to be possible in dimensions d \geq 4-- the walk gets trapped for time of order n in a small spatial region. This proves that the strategy used to infer subdiffusive lower bounds on the heat kernel in earlier studies of this problem is in fact dominant. In addition, we settle a conjecture on the maximal possible subdiffusive decay in four dimensions and prove that anomalous decay is a tail and thus zero-one event. Joint work with Marek Biskup, Alexander Vandenberg and Alexander Rozinov.

L-moments are expectations of certain linear combinations of order statistics. They form the basis of a general theory which covers the summarization and description of theoretical probability distributions, the summarization and description of observed data samples, estimation of parameters and quantiles of probability distributions, and hypothesis tests for probability distributions. L-moments are in analogous to the conventional moments, but are more robust to outliers in the data and enable more secure inferences to be made from small samples about an underlying probability distribution. They can be used for estimation of parametric distributions, and can sometimes yield more efficient parameter estimates than the maximum-likelihood estimates. This talk gives a general summary of L-moment theory and methods, describes some applications ranging from environmental data analysis to financial risk management, and indicates some recent developments on nonparametric quantile estimation, "trimmed" L-moments for very heavy-tailed distributions, and L-moments for multivariate distributions.

We review various classifications of probability distributions based on their tail heaviness. Using a characterization of medium-tailed distributions we propose a test for testing the null hypothesis of medium-tail vs long- or short-tailed distributions. Some operating characteristics of the proposed test are discussed.

Excursion sets of stationary random fields have attracted much attention in recent years.They have been applied to modeling complex geometrical structures in tomography, astro-physics and hydrodynamics. Given a random field and a specified level, it is natural to studygeometrical functionals of excursion sets considered in some bounded observation window.Main examples of such functionals are the volume, the surface area and the Euler charac-teristics. Starting from the classical Rice formula (1945), many results concerning calculationof moments of these geometrical functionals have been proven. There are much less resultsconcerning the asymptotic behavior (as the window size grows to infinity), as random variablesconsidered here depend non-smoothly on the realizations of the random field. In the talk wediscuss several recent achievements in this domain, concentrating on asymptotic normality andfunctional central limit theorems.