Georgia Institute of Technology College of Sciences

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.

The prevalence of rare events accompanied by disastrous economic and social consequences, the so-called Black-Swan events, makes today's world far different from just decades ago. In this talk, I shall address the issue of modeling the wealth process of an insurer in a stochastic economic environment with dependent insurance and financial risks. The asymptotic behavior of the finite-time ruin probability will be studied. As an application, I shall discuss a portfolio optimization problem. This talk is based on recent joint works with Raluca Vernic and Zhongyi Yuan.

The two weight inequality for the Hilbert transform arises in the settings of analytic function spaces, operator theory, and spectral theory, and what would be most useful is a characterization in the simplest real-variable terms. We show that the $L^2$ to $L^2$ inequality holds if and only if two $L^2$ to weak-$L^2$ inequalities hold. This is a corollary to a characterization in terms of a two-weight Poisson inequality, and a pair of testing inequalities on bounded functions. Joint work with Eric Sawyer, Chun-Yun Shen, and Ignacio Uriate-Tuero.

Chemical reaction networks taken with mass-action kinetics are dynamical systems governed by polynomial differential equations that arise in systems biology. In general, establishing the existence of (multiple) steady states is challenging, as it requires the solution of a large system of polynomials with unknown coefficients. If, however, the steady state ideal of the system is a binomial ideal, then we show that these questions can be answered easily. This talk focuses on systems with this property, are we say such systems have toric steady states. Our main result gives sufficient conditions for a chemical reaction system to admit toric steady states. Furthermore, we analyze the capacity of such a system to exhibit multiple steady states. An important application concerns the biochemical reaction networks networks that describe the multisite phosphorylation of a protein by a kinase/phosphatase pair in a sequential and distributive mechanism. No prior knowledge of chemical reaction network theory or binomial ideals will be assumed. (This is joint work with Carsten Conradi, Mercedes P\'erez Mill\'an, and Alicia Dickenstein.)