Free probability is maybe the most popular part of non-commutative probability theory. This young field, at the intersection of probability theory, operator algebra, complex analysis and combinatorics is also connected to other sciences, from physics to psychology and engineering. But freeness is not the only notion of non-commutative independence. Under certain universality assumptions one gets 2,3, or 4 types of independence and various interpolation models. In this talk I will give an introduction to non-commutative probability , some of its specific techniques and problems it has tackled I will emphasize the combinatorial part, where root systems, partitions, crossings, embracings, planar and rooted trees aggregate in an effective machinery.
I will talk about statistical inference of trends in mean non-stationary models, and mean regression and conditional variance (or volatility) functions in nonlinear stochastic regression models. Simultaneous confidence bands are constructed and the coverage probabilities are shown to be asymptotically correct. The Simultaneous confidence bands are useful for model specification problems in nonlinear time series. The results are applied to environmental and financial time series.
I will discuss estimation of covariance matrices of stationary processes. Under a short-range dependence condition for a wide class of nonlinear processes, I will show that the banded covariance matrix estimates converge in operator norm to the true covariance matrix with explicit rates of convergence. I will also consider the consistency of the estimate of the inverse covariance matrix. These results are applied to a prediction problem, and error bounds for the finite predictor coefficients are obtained. The work is joint with Mohsen Pourahmadi of TAMU.
Harnack inequalities describe, in quantitative ways, behavior of solutions to partial differential equations. These inequalities were originally defined for harmonic functions in the plane and much later became an important tool in the theory of harmonic functions and, more generally, partial differential equations. The purpose of my talk is to explain major ideas behind Harnack inequalities in different cases. The emphasis is in the qualitative behavior of solutions. Moreover, I will introduce a few consequences that may be deduced from Harnack inequalities, motivating the study of them. They are both deep and powerful. The understanding of Harnack inequalities for solutions to a general class of nonlinear parabolic PDEs has risen significantly recently. I will explain the history of the problem, reviewing fundamental works of De Giorgi and Moser in the linear case, and then introducing new results for a general class of equations with degenerate structure. I will also very briefly introduce main techniques to prove Harnack inequalities in different cases.
Abstract: I will present an asymptotic expansion for probabilities of moderate deviations for iid random variables and for stationary processes. The sharpness of moment conditions will be discussed. The dependence measures are easily verifiable (cf W. B. Wu (2005), Nonlinear system theory: Another look at dependence. Proc Natl Acad Sci USA. 102)
I will discuss Fourier and wavelet transforms of stationary, causal processes. Under mild conditions, Fourier transforms are shown to be asymptotically independent complex Gaussian at different frequencies. To this end, I will apply Carleson's Theorem, a very deep result in harmonic analysis.
It is shown that the McKay (1993) and Jones, McKay and Hu (1994) modifications of Abramson's (1982) variable bandwidth kernel density estimator satisfies optimal asymptotic properties for estimating densities with four or six uniformly continuous derivatives, uniformly on bounded sets where the preliminary estimator of the density is bounded away from zero.
Abstract: I will present an asymptotic expansion for probabilities of moderate deviations for iid random The sharpness of moment conditions will be discussed. The dependence measures are easily verifiable (cf W. B. Wu (2005), Nonlinear system theory: Another look at dependence. Proc Natl Acad Sci USA. 102)
The Tracy-Widom functions describe the limiting distribution of a variety of statistical quantities, including the largest eigenvalue of a random matrix drawn from the Gaussian orthogonal, symplectic, or unitary ensembles (GOE, GSE, or GUE), the longest increasing subsequence of a random permutation, and the outermost particle in a sea of non-intersecting Brownian particles. We obtain new formulas for the Tracy-Widom functions in terms of integrals of Painleve functions. Using these new formulas we find the complete asymptotic expansion of the left-hand tail of the GOE and GSE Tracy-Widom functions for the first time, as well as a second proof of the recently obtained result for the GUE case. We conclude by discussing progress on a new family of "incomplete" Tracy-Widom distributions corresponding to the largest observed eigenvalue if each eigenvalue has a fixed probability of being observed. This is joint work with Jinho Baik and Jeffery DiFranco.
We introduce a measure on strict plane partitions that is an analog of the uniform measure on plane partitions. We describe this measure in terms of a Pfaffian point process and compute its bulk limit when partitions become large.The above measure is a special case of the shifted Schur process, which generalizes the shifted Schur measure introduced by Tracy and Widom and is an analog of the Schur process introduced by Okounkov and Reshetikhin. We use the Fock space formalism to prove that the shifted Schur process is a Pfaffian point process and calculate its correlation kernel.
We also obtain a generalization of MacMahon's formula for the generating function of plane partitions. We give a 2-parameter generalization related to Macdonald's symmetric functions. The formula is especially simple in the Hall-Littlewood case.
Limit theorems for a hyperbolic or partially hyperbolic dynamical system are usually proved by means of a clever partitioning the phase space of the system. This should lead to the creation of a family of sigma-filelds with customary mixing properties when the machinery of weak dependence is applicable. We are going to consider an alternative approach when no cutting of the phase space is performed. Instead, by means of probabilistic tools an extension of the original dynamical system is constructed supplied with a family of sigma-field enjoying nice mixing properties. We will discuss advantages and drawbacks of this approach and consider ergodic toral automorphisms as examples where this approach goes smoothly and leads to new conclusions.
An interesting estimation problem, arising in many dynamical systems, is that of filtering; namely, one wishes to estimate a trajectory of a signal process (which is not observed) from a given path of an observation process, where the latter is a nonlinear functional of the signal plus noise.In the classical mathematical framework, the stochastic processes are parameterized by a single parameter (interpreted as ``time''), the observation noise is a martingale (say, a Brownian motion), and the best mean-square estimate of the signal, called the optimal filter, has a number of useful representations and satisfies the well-known Kushner-FKK and Duncan-Mortensen-Zakai stochastic partial differential equations.
However, there are many applications, arising, for example, in connection with denoising and filtering of images and video-streams, where the parameter space has to be multidimensional. Another level of difficulty is added if the observation noise has a long-memory structure, which leads to nonstandard filtering evolution equations. Each of the two features (multidimensional parameter space and long-memory observation noise) does not permit the use of the classical theory of filtering and the combination of the two has not been previously explored in mathematical literature on stochastic filtering.
This talk focuses on nonlinear filtering of a signal in the presence of long-memory fractional Gaussian noise. We will start by introducing first the evolution equations and integral representations of the optimal filter in the one-parameter case, when the noise driving the observation is represented by a fractional Brownian motion. Next, using fractional calculus and multiparameter martingale theory, the case of spatial nonlinear filtering of a random field observed in the presence of a persistent fractional Brownian sheet will be explored.
Abstract: Permanents of random matrices with iid entries converge to lognormal or normal variables. For generalized permanents the limit is described in terms of the multiple Ito-Wiener integral of elements of Hoeffding decomposition. This theory parallels the one which has been developed for U-statistics. Applications for counting problems for perfect matchings in bi-partite graphs include for instance counting monochromatic matchings or counting matchings with a given color structure of edges. The talk is based on Ch. 5 of a little book "Symmetric Functionals on Random Matrices and Random Matching Problems" (Springer, 2008) co-authored by Grzegorz Rempala and myself.