Purdue Probability Seminar

# Маленький семинар по теории вероятности Fall 2018

This semester we are experimenting a short Russian seminar format: talks are 50mn long and slides are forbidden.

Wednesdays in UNIV 101 from 1:30-2:20pm, unless otherwise noted.

Directions to Purdue by air or by car.

Schedule of talks from Spring 2018, Fall 2017, Spring 2017 and Fall 2016.

### Title

8/29/18 Anirban DasGupta
Purdue University
On Converting Hardy-Landau-Kolmogorov variational inequalities to variational diameters of classes of measures
Abstract
In analysis, a typical inequality of the Hardy-Landau-Kolmogorov type connects, and usually upper bounds, norms of intermediate derivatives of sufficiently smooth functions on the real line or a segment of it by using norms of higher order derivatives. The inequalities usually come with universal constants in front, and in some cases, mostly because of Kolmogorov's work, the best constant is known, at least in principle.
What do these have to do with statistics? We will describe in this seminar how to combine these inequalities with other inequalities in other domains of analysis to ultimately produce bounds, explicit, on the total variation diameters of what statisticians call shape-restricted non-parametric densities. For example, the shape restriction could be monotonicity, or convexity, or logconcavity. The inequalities that the Hardy-Landau-Kolmorov inequalities are combined with are convexity inequalities. For example, Young's inequality, or Ostrowski's inequality, etc.
Are there any practical uses? Yes, powers of likelihood ratio tests, or, local limit theorems.
9/5/18 Vivian Healey
University of Chicago
Tree Embedding via the Generalized Loewner Equation
Abstract
In its most well-known form, the Loewner equation gives a correspondence between curves in the upper half-plane and continuous real functions (called the 'driving function' for the equation). We consider the generalized Loewner equation, where the driving function has been replaced by a time-dependent real measure. In the first part of the talk, we investigate the delicate relationship between the driving measure and the generated hull, specifying a class of discrete random driving measures that generate hulls in the upper half-plane that are embeddings of trees. In the second part of the talk, we consider the probabilistic question of finding the scaling limit of these measures for Galton-Watons trees. We particularly focus on distributions on trees converging to the continuum random tree, and we conclude by describing connections to the complex Burgers equation. Joint work with Govind Menon (Brown University).
9/12/18 Andrew Thomas
Purdue University
Limit theorems for process-level Betti numbers for sparse, critical and Poisson regimes
Abstract
The objective of this study is to examine the asymptotic behavior of Betti numbers of Čech complexes treated as stochastic processes and formed from random points in the d-dimensional Euclidean space ${\scriptsize R^{d}}$. We consider the case where the points of the Čech complex are generated by a Poisson process with intensity ${\scriptsize n_f}$ for a probability density ${\scriptsize f}$. We look at the cases where the behavior of the connectivity radius of Čech complex causes simplices of dimension greater than ${\scriptsize k + 1}$ to vanish in probability, the so-called sparse and Poisson regimes, as well when the connectivity radius is on the order of ${\scriptsize n−1/d}$, the critical regime. We establish limit theorems in all of the aforementioned regimes, a central limit theorem for the sparse and critical regimes, and a Poisson limit theorem for the Poisson regime. When the connectivity radius of the Čech complex is ${\scriptsize o(n−1/d)}$, i.e., the sparse and Poisson regimes, we can decompose the limiting processes into a time-changed Brownian motion and a time-changed homogeneous Poisson process respectively. In the critical regime, the limiting process is a centered Gaussian process but has much more complicated representation, because the Čech complex becomes highly connected with many topological holes of any dimension. This is joint work with Takashi Owada (Purdue).
9/19/18 Phanuel Mariano
Purdue University
Coupling hypoelliptic diffusions and applications to gradient estimates
Abstract
Coupling is a way of constructing Markov processes with prescribed laws on the same space. In 1991, M. Cranston was the first to show the connection between coupling and analytic gradient estimates. We will start by introducing the notion of coupling and review the known results for the elliptic setting. In this talk, we consider coupling of two hypoelliptic diffusions (diffusions driven by vector fields whose Lie algebra span the whole tangent space). We construct a successful non-Markovian coupling on the Heisenberg group, which is the first example of a nontrivial sub-Riemannian manifold. We use this coupling to obtain gradient estimates similar to the one’s Cranston obtained on Riemannian manifolds. We will also obtain similar results for the degenerate Kolmogorov diffusion. This talk is based on joint work with Sayan Banerjee and Maria Gordina.
9/26/18 Jing Wang
Purdue University
TBA
Abstract
Abstract coming soon.
10/3/18 Xia Chen
University of Tennessee
TBA
Abstract
Abstract coming soon.
10/17/18 Robin Pemantle
University of Pennsylvania
TBA
Abstract
Abstract coming soon.
UIUC
TBA
Abstract
Abstract coming soon.
10/31/18 Jon Peterson
Purdue University
TBA
Abstract
Abstract coming soon.
11/7/18 Eviatar Proccacia
Texas A&M
TBA
Abstract
Abstract coming soon.
11/14/18 Gautam Iyer
Carnegie Mellon University
TBA
Abstract
Abstract coming soon.
11/28/18 Prakash Chakraborty
Purdue University
TBA
Abstract
Abstract coming soon.
12/5/18 Gugan Thoppe
Duke University
TBA
Abstract
Abstract coming soon.