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Series: Stochastics Seminar

Series: Stochastics Seminar

Series: Stochastics Seminar

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).

Series: Stochastics Seminar

Series: Stochastics Seminar

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.

Series: Stochastics Seminar

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

Series: Stochastics Seminar

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.

Series: Stochastics Seminar

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.

Series: Stochastics Seminar

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.

Series: Stochastics Seminar

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.