Prevalence of heavy-tailed distributions in systems with multiple scales: insights through stochastic averagingThursday, October 18, 2018 - 15:05 , Location: Skiles 006 , Rachel Kuske , School of Mathematics, Georgia Tech , Organizer: Christian Houdre
Heavy tailed distributions have been shown to be consistent with data in a variety of systems with multiple time scales. Recently, increasing attention has appeared in different phenomena related to climate. For example, correlated additive and multiplicative (CAM) Gaussian noise, with infinite variance or heavy tails in certain parameter regimes, has received increased attention in the context of atmosphere and ocean dynamics. We discuss how CAM noise can appear generically in many reduced models. Then we show how reduced models for systems driven by fast linear CAM noise processes can be connected with the stochastic averaging for multiple scales systems driven by alpha-stable processes. We identify the conditions under which the approximation of a CAM noise process is valid in the averaged system, and illustrate methods using effectively equivalent fast, infinite-variance processes. These applications motivate new stochastic averaging results for systems with fast processes driven by heavy-tailed noise. We develop these results for the case of alpha-stable noise, and discuss open problems for identifying appropriate heavy tailed distributions for these multiple scale systems. This is joint work with Prof. Adam Monahan (U Victoria) and Dr. Will Thompson (UBC/NMi Metrology and Gaming).
Thursday, September 20, 2018 - 15:05 , Location: Skiles 006 , Konstantin Tikhomirov , School of Mathematics, GaTech , Organizer: Christian Houdre
Let (A_n) be a sequence of random matrices, such that for every n, A_n is n by n with i.i.d. entries, and each entry is of the form b*x, where b is a Bernoulli random variable with probability of success p_n, and x is an independent random variable of unit variance. We show that, as long as n*p_n converges to infinity, the appropriately rescaled spectral distribution of A_n converges to the uniform measure on the unit disc of complex plane. Based on joint work with Mark Rudelson.
Thursday, September 6, 2018 - 15:05 , Location: Skiles 006 , Sara van de Geer , ETH Zurich , Organizer: Mayya Zhilova
The seminar will be the third lecture of the TRIAD Distinguished Lecture Series by Prof. Sara van de Geer. For further information please see http://math.gatech.edu/events/triad-distinguished-lecture-series-sara-van-de-geer-0
Thursday, August 30, 2018 - 15:05 , Location: Skiles 006 , Andrew Nobel , University of North Carolina, Chapel Hill , Organizer: Mayya Zhilova
This talk concerns the description and analysis of a variational framework for empirical risk minimization. In its most general form the framework concerns a two-stage estimation procedure in which (i) the trajectory of an observed (but unknown) dynamical system is fit to a trajectory from a known reference dynamical system by minimizing average per-state loss, and (ii) a parameter estimate is obtained from the initial state of the best fit reference trajectory. I will show that the empirical risk of the best fit trajectory converges almost surely to a constant that can be expressed in variational form as the minimal expected loss over dynamically invariant couplings (joinings) of the observed and reference systems. Moreover, the family of joinings minimizing the expected loss fully characterizes the asymptotic behavior of the estimated parameters. I will illustrate the breadth of the variational framework through applications to the well-studied problems of maximum likelihood estimation and non-linear regression, as well as the analysis of system identification from quantized trajectories subject to noise, a problem in which the models themselves exhibit dynamical behavior across time.
Tuesday, June 12, 2018 - 15:05 , Location: Skiles 006 , Jean-Christophe Breton , University of Rennes , Organizer: Mayya Zhilova
Random balls models are collections of Euclidean balls whose centers and radii are generated by a Poisson point process. Such collections model various contexts ranging from imaging to communication network. When the distributions driving the centers and the radii are heavy-tailed, interesting interference phenomena occurs when the model is properly zoomed-out. The talk aims to illustrate such phenomena and to give an overview of the asymptotic behavior of functionals of interest. The limits obtained include in particular stable fields, (fractional) Gaussian fields and Poissonian bridges. Related questions will also be discussed.
Thursday, April 19, 2018 - 15:05 , Location: Skiles 006 , Tomasz Tkocz , Carnegie Mellon University , firstname.lastname@example.org , Organizer: Michael Damron
We shall prove that a certain stochastic ordering defined in terms of convex symmetric sets is inherited by sums of independent symmetric random vectors. Joint work with W. Bednorz.
Thursday, April 12, 2018 - 15:05 , Location: Skiles 006 , Joshua Rosenberg , University of Pennsylvania , email@example.com , Organizer: Michael Damron
In this talk I will explore the subject of Bernoulli percolation on Galton-Watson trees. Letting $g(T,p)$ represent the probability a tree $T$ survives Bernoulli percolation with parameter $p$, we establish several results relating to the behavior of $g$ in the supercritical region. These include an expression for the right derivative of $g$ at criticality in terms of the martingale limit of $T$, a proof that $g$ is infinitely continuously differentiable in the supercritical region, and a proof that $g'$ extends continuously to the boundary of the supercritical region. Allowing for some mild moment constraints on the offspring distribution, each of these results is shown to hold for almost surely every Galton-Watson tree. This is based on joint work with Marcus Michelen and Robin Pemantle.