Seminars and Colloquia by Series

Some new non-asymptotic results about the accuracy of the weighted bootstrap

Series
Stochastics Seminar
Time
Thursday, April 28, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mayya Zhilova – School of Mathematics, Georgia Tech
The bootstrap procedure is well known for its good finite-sample performance, though the majority of the present results about its accuracy are asymptotic. I will study the accuracy of the weighted (or multiplier) bootstrap procedure for estimation of quantiles of a likelihood ratio statistic. The set-up is the following: the sample size is bounded, random observations are independent, but not necessarily identically distributed, and a parametric model can be misspecified. This problem had been considered in the recent work of Spokoiny and Zhilova (2015) with non-optimal results. I will present a new approach improving the existing results.

Levy-Khintchine random matrices and the Poisson weighted infinite skeleton tree

Series
Stochastics Seminar
Time
Thursday, April 21, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Paul Jung – University of Alabama Birmingham
We look at a class of Hermitian random matrices which includes Wigner matrices, heavy-tailed random matrices, and sparse random matrices such as adjacency matrices of Erdos-Renyi graphs with p=1/N. Our matrices have real entries which are i.i.d. up to symmetry. The distribution of entries depends on N, and we require sums of rows to converge in distribution; it is then well-known that the limit must be infinitely divisible. We show that a limiting empirical spectral distribution (LSD) exists, and via local weak convergence of associated graphs, the LSD corresponds to the spectral measure associated to the root of a graph which is formed by connecting infinitely many Poisson weighted infinite trees using a backbone structure of special edges. One example covered are matrices with i.i.d. entries having infinite second moments, but normalized to be in the Gaussian domain of attraction. In this case, the LSD is a semi-circle law.

Dynamics of geodesic flows with random forcing on Lie groups with left-invariant metrics

Series
Stochastics Seminar
Time
Thursday, April 7, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Wenqing Hu – University of Minnesota, Twin Cities
Motivated by problems in turbulent mixing, we consider stochastic perturbations of geodesic flow for left-invariant metrics on finite-dimensional Lie groups. We study the ergodic properties and provide criteria that ensure the Hormander condition for the corresponding Markov processes on phase space. Two different types of models are considered: the first one is a classical Langevin type perturbation and the second one is a perturbation by a “conservative noise”. We also study an example of a non-compact group. Joint work with Vladimir Sverak.

Random walks on abelian sandpiles

Series
Stochastics Seminar
Time
Thursday, March 31, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
John Pike – Cornell University
Given a simple connected graph G=(V,E), the abelian sandpile Markov chain evolves by adding chips to random vertices and then stabilizing according to certain toppling rules. The recurrent states form an abelian group \Gamma, the sandpile group of G. I will discuss joint work with Dan Jerison and Lionel Levine in which we characterize the eigenvalues and eigenfunctions of the chain restricted to \Gamma in terms of "multiplicative harmonic functions'' on V. We show that the moduli of the eigenvalues are determined up to a constant factor by the lengths of vectors in an appropriate dual Laplacian lattice and use this observation to bound the mixing time of the sandpile chain in terms of the number of vertices and maximum vertex degree of G. We also derive a surprising inverse relationship between the spectral gap of the sandpile chain and that of simple random walk on G.

Randomized Isoperimetric Inequalities

Series
Stochastics Seminar
Time
Tuesday, March 8, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Peter Pivovarov – University of Missouri
The focus of my talk will be stochastic forms of isoperimetric inequalities for convex sets. I will review some fundamental inequalities including the classical isoperimetric inequality and those of Brunn-Minkowski and Blaschke-Santalo on the product of volumes of a convex body and its polar dual. I will show how one can view these as global inequalities that arise via random approximation procedures in which stochastic dominance holds at each stage. By laws of large numbers, these randomized versions recover the classical inequalities. I will discuss when such stochastic dominance arises and its applications in convex geometry and probability. The talk will be expository and based on several joint works with G. Paouris, D. Cordero-Erausquin, M. Fradelizi, S. Dann and G. Livshyts.

On the analogue of the monotonicity of entropy in the Brunn-Minkowski theory

Series
Stochastics Seminar
Time
Thursday, March 3, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Arnaud Marsiglietti – IMA, University of Minnesota
In the late 80's, several relationships have been established between the Information Theory and Convex Geometry, notably through the pioneering work of Costa, Cover, Dembo and Thomas. In this talk, we will focus on one particular relationship. More precisely, we will focus on the following conjecture of Bobkov, Madiman, and Wang (2011), seen as the analogue of the monotonicity of entropy in the Brunn-Minkowski theory: The inequality $$ |A_1 + \cdots + A_k|^{1/n} \geq \frac{1}{k-1} \sum_{i=1}^k |\sum_{j \in \{1, \dots, k\} \setminus \{i\}} A_j |^{1/n}, $$ holds for every compact sets $A_1, \dots, A_k \subset \mathbb{R}^n$. Here, $|\cdot|$ denotes Lebesgue measure in $\mathbb{R}^n$ and $A + B = \{a+b : a \in A, b \in B \}$ denotes the Minkowski sum of $A$ and $B$. (Based on a joint work with M. Fradelizi, M. Madiman, and A. Zvavitch.)

Central Limit Theorems and Bootstrap in High Dimensions

Series
Stochastics Seminar
Time
Thursday, February 25, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Victor Chernozhukov – MIT

Please Note: Paper available on arXiv:1412.3661

This paper derives central limit and bootstrap theorems for probabilities that sums of centered high-dimensional random vectors hit hyperrectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for probabilities Pr(n−1/2∑ni=1Xi∈A) where X1,…,Xn are independent random vectors in ℝp and Ais a hyperrectangle, or, more generally, a sparsely convex set, and show that the approximation error converges to zero even if p=pn→∞ as n→∞ and p≫n; in particular, p can be as large as O(eCnc) for some constants c,C>0. The result holds uniformly over all hyperrectangles, or more generally, sparsely convex sets, and does not require any restriction on the correlation structure among coordinates of Xi. Sparsely convex sets are sets that can be represented as intersections of many convex sets whose indicator functions depend only on a small subset of their arguments, with hyperrectangles being a special case.

On the infinitesimal versions of Log Brunn Minkowski and Gaussian Brunn Minkowski conjectures

Series
Stochastics Seminar
Time
Thursday, February 18, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Galyna Livshyts – School of Mathematics, Georgia Tech
Log Brunn-Minkowski conjecture was proposed by Boroczky, Lutwak, Yang and Zhang in 2013. It states that in the case of symmetric convex sets the classical Brunn-MInkowski inequality may be improved. The Gaussian Brunn-MInkowski inequality was proposed by Gardner and Zvavitch in 2007. It states that for the standard Gaussian measure an inequality analogous to the additive form of Brunn_minkowski inequality holds true for symmetric convex sets. In this talk we shall discuss a derivation of an equivalent infinitesimal versions of these inequalities for rotation invariant measures and a few partial results related to both of them as well as to the classical Alexander-Fenchel inequality.

Random zero sets under repeated differentiation of an analytic function

Series
Stochastics Seminar
Time
Thursday, February 4, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sneha Subramanian – School of Mathematics, Georgia Tech
For a random (complex) entire function, what can we say about the behavior of the zero set of its N-th derivative, as N goes to infinity? In this talk, we shall discuss the result of repeatedly differentiating a certain class of random entire functions whose zeros are the points of a Poisson process of intensity 1 on the real line. We shall also discuss the asymptotic behavior of the coefficients of these entire functions. Based on joint work with Robin Pemantle.

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