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Series: Stochastics Seminar

I will discuss the limit theorems for composition of analytic functions on the upper-half-plane, and the analogies and differences with the limit theorems for sums of independent random variables. The analogies are enhanced by recalling that the probabilistic limit theorems are really results about convolution of probability measures, and by introducing a new binary operation on probability measures, the monotone convolution.This is joint work with John D. Williams.

Series: Stochastics Seminar

Ergodic theory of randomly forced space-time homogeneous Burgers
equation in noncompact setting has been developed in a recent paper by
Eric Cator , Kostya Khanin, and myself. The analysis is based on first
passage percolation methods that allow to study coalescing one-sided
action minimizers and construct the global solution via Busemann
functions. i will talk about this theory and its extension to the case
of space-continuous kick forcing. In this setting, the minimizers do
not coalesce, so for the ergodic program to go through, one must use
new soft results on their behavior to define generalized Busemann
functions along appropriate subsequences.

Series: Stochastics Seminar

We define the class of ultra sub-Gaussian random vectors and
derive optimal comparison of even moments of linear combinations of such
vectors in the case of the Euclidean norm. In particular, we get optimal
constants in the classical Khinchine inequality. This is a joint work with
Krzysztof Oleszkiewicz.

Series: Stochastics Seminar

In this talk we will consider a finite sample of i.i.d. random variables which are uniformly distributed in some convex body in R^d. We will propose several estimators of the support, depending on the information that is available about this set: for instance, it may be a polytope, with known or unknown number of vertices. These estimators will be studied in a minimax setup, and minimax rates of convergence will be given.

Series: Stochastics Seminar

The CLT for free random variables was settled by Voiculescu very early in this work on free probability. He used this in turn to prove his main result on aymptotic freeness of independent random matrices. On the other hand, in random matrices, fluctuations can be understood as a second order phenomena. This notion of fluctuations has a conterpart in free probability which is called freenes of second order. I will explain what this is and how one can prove a free CLT result in this context. It is also interesting to point out that this is a nontrivial calculation which begs the same question in the classical context and I will comment on that.

Series: Stochastics Seminar

We will discuss sharp bounds on moments and concentration inequalities for
the operator norm of deviations of sample covariance operators from the
true covariance operator for i.i.d. Gaussian random variables in a
separable Banach space.
Based on a joint work with Karim Lounici.

Series: Stochastics Seminar

Let (X_k)_{k \geq 1} and (Y_k)_{k\geq1} be two independent
sequences of independent identically distributed random variables
having the same law and taking their values in a finite alphabet
\mathcal{A}_m. Let LC_n be the length of the longest common
subsequence of the random words X_1\cdots X_n and Y_1\cdots Y_n.
Under assumptions on the distribution of X_1, LC_n is shown to
satisfy a central limit theorem. This is in contrast to the Bernoulli
matching problem or to the random permutations case, where the limiting
law is the Tracy-Widom one. (Joint with Umit Islak)

Series: Stochastics Seminar

A sandpile on a graph is an integer-valued function on the vertices. It evolves according to local moves called topplings. Some sandpiles stabilize after a finite number of topplings, while others topple forever. For any sandpile s_0 if we repeatedly add a grain of sand at an independent random vertex, we eventually reach a sandpile s_\tau that topples forever. Statistical physicists Poghosyan, Poghosyan, Priezzhev and Ruelle conjectured a precise value for the expected amount of sand in this "threshold state" s_\tau in the limit as s_0 goes to negative infinity. I will outline the proof of this conjecture in http://arxiv.org/abs/1402.3283 and explain the big-picture motivation, which is to give more predictive power to the theory of "self-organized criticality".

Series: Stochastics Seminar

In this talk we investigate possible applications of the infinitedimensional Gaussian Radon transform for Banach spaces to machine learning. Specifically, we show that the Gaussian Radon transform offers a
valid stochastic interpretation to the ridge regression problem in the case when the reproducing kernel Hilbert space in question is infinite-dimensional. The main idea is to work with stochastic processes defined not on the Hilbert
space itself, but on the abstract Wiener space obtained by completing the Hilbert space with respect to a measurable norm.

Series: Stochastics Seminar

Let $Y$ be a nonnegative random variable with mean $\mu$, and let $Y^s$, defined on the same space as $Y$, have the $Y$ size biased distribution, that is, the distribution characterized by $\mathbb{E}[Yf(Y)]=\mu \mathbb{E}[f(Y^s)]$ for all functions $f$ for which these expectations exist. Under bounded coupling conditions, such as $Y^s-Y \leq C$ for some $C>0$, we show that $Y$ satisfies certain concentration inequalities around $\mu$. Examples will focus on occupancy models with log-concave marginal distributions.