This talk will survey some distances between two probability laws and the relations between them. They are used to obtain various rates of convergence in the central limit theorem for sums of independent and dependent random variables.
The logarithmic Sobolev inequality, discovered by L. Gross, is a weak form of a Sobolev inequality: it gives control of a function through average information about its derivatives. It has become a ubiquitous tool in global analysis and probability theory, with important applications in stochastic analysis, large deviations, spectral theory, nonlinear PDE, geometric analysis, noncommutative geometry, and more. As a stunning example, the logarithmic Sobolev inequality inspired Perelman's proof of the Poincar\'e conjecture.In this talk, I will give an introduction to the area of logarithmic Sobolev inequalities. I will then discuss new insights on such inequalities in the context of regular function spaces, such as holomorphic and subharmonic functions. In recent work with P. Graczyk, J. Loeb, and T. Zak, we have discovered a new and very general logarithmic Sobolev inequality for logarithmicallysubharmonic functions. I will discuss these results and give clues about their significance.

Stochastic Calculus study grouporganized by B. Zhang and T. Oraby will study An Introduction to Stochastic Differential Equations (version 1.2) by L. C. Evans.Tuesdays 57 in the math launge

Abstract: The Law of Large Numbers, the Central Limit Theorem, and the Law of the Iterated Logarithm for independent and identically distributed sequences of random variables are three central, perhaps dominant, results of classical probability theory. The Ergodic Theorem provides a complete extension of the Law of Large Numbers to sequences that are dependent, but stationary. The Central Limit Theorem and Law of the Iterated Logarithm do not extend as completely, but only under additional conditions that effectively limit the amount of dependence. During the past decade there has been some progress on understanding the Central Limit Theorem for stationary processes, resulting in conditions that are sufficient and nearly necessary, at least for the conditional version of the Central Limit Theorem. The talk will present recent efforts to modify the arguments leading to the Central Limit Theorem to obtain a Law of the Iterated Logarithm. It will begin with some background material on the Law of the Iterated Logarithm and a selective review of recent work on the Central Limit Theorem for stationary sequences. It will then describe the modifications necessary to obtain the Law of the Iterated Logarithm.
Abstract: In this joint work with J. Dedecker and E. Rio, we obtain convergence rates in the central limit theorem for stationary sequences in L_{p} for Wasserstein distances of order r, for p in ]2,3] and r in ]p2,p]. The conditions are expressed in terms of projective criteria. The results apply in particular to nonadapted sequences.
Informally, the simple exclusion process follows a collection of random walks which interact in that they are not allowed to jump onto each other. In this talk, we consider the motion of a distinguished, or tagged, particle in this particle system. We review some of the past results and discuss some new contributions.
 Functional Analysis SeminarWednesday 34 pm, Feb 21, Feb 28, March 7 Seminar room (OC 807) Victor Kaftal, Majorization theory for infinite sequencesMajorization for finite sequences is linked to doubly stochastic matrices, convexity, the diagonals of selfadjoint matrices (the SchurHorn Theorem) and more, and so it has been of interest to researchers in several areas of math. Little was know until recently about majorization for infinite sequences but we have now some new results. 
We revisit some cumulant methods for Independent Component Analysis  an unsupervised learning method with increasing popularity and applicability in number of disciplines. A rigorous justification of identifiability of the linear ICA method by kurtosis maximization is given by a simple lemma from optimization theory, which gives a basis for a generalization of the famous Fixed Point Algorithm for ICA for high order cumulants. We propose a measure for independence of group of random variables, given by a sum of crosscumulants of a given order n. Similar measure was known for the case of four order crosscumulants from the JADE algorithm for ICA. We derive a formula for its calculation using cumulant tensors. In the case n=4 our formula allows efficient calculation of this measure, using cumulant matrices. Much attention is devoted to the case of six order crosscumulants, aiming to show that this measure can be calculated using again cumulant matrices. We provide a simple proof of the main ICA theorem concerning identifiability of the linear ICA model using the properties of the crosscumulants instead of the DarmoisSkitovitch theorem from statistics, used for this purpose in the literature on ICA. Various ICA algorithms are demonstrated.
I shall present some results about the central limit theorem and its weak invariance principle for sums of nonadapted stationary sequences, under different normalizations. Our conditions involve the conditional expectation of the variables with respect to the given sigmaalgebra, as done in Gordin (1969) and Heyde (1974.) These conditions are welladapted to a large variety of examples, including linear processes with dependent innovations or regular functions of linear processes.
The "zerorange" system is an (infinite) collection of dependent random walks on Z^{d} which models various types of traffic. Informally, the interaction is in that a particle jumps with a rate depending on the number of particles at its vertex, but to where it jumps is selected independently. In this talk, we consider a distinguished, or "tagged," particle in this system and discuss its asymptotic behavior including some recent diffusive estimates in "equilibrium." In particular cases, we also discuss approximation of the tagged position by a Brownian motion with parameters depending on the form of the interaction and the density of particles.
This is a "practice talk for a conference". The topic is classificiatioon of rpocesses with linear regressions and quadratic conditional variances.
Schedule

Related Activities

Statistics SeminarWednesdays 34, Room 825 
Abstract We survey the role played by orthogonal polynomials in the analysis of transision probabilities of birth and death processes and associated random walks. Several examples will be mentioned.
Abstract Stieltjes and Hilbert computed discriminants of Jacobi polynomials. Stieltjes studied the electrostatics equilibrium problem of nunit charged particles restricted to (1,1) under the external field of charges (a+1)/2 and (b+1)/2 at ±1. The potential is a logarithmic potential. Stieljes showed that the equilibrium position of the particle is at the zeros of the Jacobi polynomial P_{n}^{(a, b)}(x). We discuss the recent developments on this problem, its extension to general orthogonal polynomials and the role discriminants play in the solution of the problem. We also mention the more recent work where similar techniques are used to solve the Bethe Ansatz equations for the XXZ and XXX models.
Abstract We assume that realworld financial markets are partially hedgable, therefore they are fundamentally incomplete [3]. It has been known for some time that there is no unique price for financial contracts in incomplete market. Recently, by the help of reduced MongeAmpere equations, [1] introduces a new method for pricing and hedging of financial instruments in in/complete market. Financial contracts still have nonunique prices in incomplete setting, but the prices depend on only g, socalled relative risk aversion.
 [1] S.D. Stojanovic: Higher dimensional fair option pricing and hedging under Hara and Cara utilities. (Preprint, August 2005)
 [2] S.D. Stojanovic: Pde methods in financial modeling. 25th Annual Searcde in Dayton
 [3] S.D. Stojanovic: Actuaries vs. Financial Engineers in regard to valuation: the truth is now found to be in between. Garp Risk Review, Sept/Oct 2005
Schedule

FA Seminar We 23 Room 708
