## Probability seminar

• Title: A LAW OF THE ITERATED LOGARITHM FOR GRENANDER'S ESTIMATOR
• Speaker: Jon Wellner (University of Washington)
• Time: 2:30 p.m., Monday, October 24, 2016
• Room: THO 325
• Abstract: I will discuss the following law of the iterated logarithm for the Grenander estimator (or MLE) $\widehat{f}_n$ of a monotone decreasing density: If $f(t_0) > 0$, $f'(t_0) < 0$, and $f'$ is continuous in a neighborhood of $t_0$, then \begin{eqnarray*} \phantom{bla} \limsup_{n\rightarrow \infty} \left ( \frac{n}{2\log \log n} \right )^{1/3} ( \widehat{f}_n (t_0 ) - f(t_0) ) = \left| f(t_0) f'(t_0)/2 \right|^{1/3} 2M \end{eqnarray*} almost surely where $$M \equiv \sup_{g \in {\cal G}} T_g = (3/4)^{1/3} \ \ \ \mbox{and} \ \ \ T_g \equiv \mbox{argmax}_u \{ g(u) - u^2 \} ;$$ here ${\cal G}$ is the two-sided Strassen limit set on $\mathbb{R}$. The proof relies on laws of the iterated logarithm for local empirical processes, Groeneboom's switching relation, and properties of Strassen's limit set analogous to distributional properties of Brownian motion. The constant in the limsup is related to the tail behavior of Chernoff's distribution, the law of $\mbox{argmax} \{W(u) - u^2\}$ where $W$ is two-sided Brownian motion starting at $0$. I will also briefly discuss several open problems connected to the asymptotic distribution of the mode estimator $M(\widehat{f}_n)$ where $\widehat{f}_n$ is the MLE of a log-concave density $f$ on $\mathbb{R}$ with $(-\log f)^{\prime \prime } (m) > 0$.

• Title: BOUNDARY TRACE OF A CLASS OF DIFFUSION PROCESSES
• Speaker: Lidan Wang (University of Washington)
• Time: 2:30 p.m., Monday, October 17, 2016
• Room: THO 325
• Abstract: For a reflecting Bessel process, the inverse local time at 0 is an $\alpha$-stable subordinator, then the corresponding subordinate Brownian motion is a symmetric $2\alpha$-stable process. Based on a discussion of Esscher and Girsanov transforms of general diffusions, we would get a comparison theorem between inverse local times of Bessel processes and perturbed Bessel processes. An immediate application would be Green function estimates of trace processes.
This is joint work with Prof. Zhen-Qing Chen.

• Title: LOCAL UNIVERSALITY OF ROOTS OF RANDOM POLYNOMIALS
• Speaker: Oanh Nguyen (Yale University)
• Time: 2:30 p.m., Monday, October 10, 2016
• Room: THO 325
• Abstract: We consider random polynomials of the form $$P_n(x) = \xi_1 p_1(x) + \xi_2 p_2(x) + \dots +\xi_n p_n(x)$$ where $\xi_1, \dots, \xi_n$ are independent random variables and $p_1, \dots, p_n$ are deterministic polynomials. Questions about the distribution of the zeros of $P_n$ have attracted intensive research for many decades with seminal papers by Kac, Littlewood-Offord, Erdos-Offord, and Tao-Vu, to name a few. In this talk, we will discuss some universality properties of the roots of generalized Kac polynomials and trigonometric random polynomials. As an application, we calculate the average number of real roots and discuss some asymptotic behavior of this number. The talk is based on some joint works with Yen Do, Hoi Nguyen, and Van Vu.

• Title: INFINITE SYSTEMS OF COMPETING BROWNIAN PARTICLES
• Speaker: Andrey Sarantsev (University of California, Santa Barbara)
• Time: 2:30 p.m., Monday, October 3, 2016
• Room: THO 325
• Abstract: We consider infinite systems (one- or two-sided) of rank-based particles on the real line. We find stable distributions and convergence results for the gaps between particles.