Prob/Stat Seminar

The Prob/Stat Seminar will meet Fridays from 11:10 to noon in Christmas-Saucon 302.


Feb. 1
Jingyu Huang (U of Birmingham, UK and Jinan University)
A central limit theorem for stochastic heat equation

We study the stochastic heat equation on the real line $$ \frac{\partial u}{\partial t} = \frac{1}{2}\frac{\partial^2 u}{\partial x^2} +\sigma(u)\dot{W}$$ (where \(\dot{W}\) is a space time white noise and \(\sigma\) is differentiable with a bounded derivative). The main result of this talk is that: the spatial integral \(\int_{-R}^R u(t, x)\, dx\) converges to a Gaussian distribution as \(R\rightarrow\infty\), after renormalization. It is proved using Stein's method and Malliavin calculus, which will be introduced in the talk. This result is based on a joint work with Nualart and Viitasaari.


Feb. 22
Berend Coster (UConn)
The financial value of weak information

Mar. 8
Elton Hsu (Northwestern)
From Geodesic Flow to Riemannian Brownian Motion

We will discuss a natural family of diffusion processes on the tangent bundle over a compact Riemannian manifold that interpolates between Brownian motion and the geodesic flow introduced by Bismut. The convergence to Brownian motion (together with an attendant Gaussian field) at one end 
of the parameter interval is nontrivial and proved in the weak sense of finite dimensional marginal distributions. By slightly extending the traditional
 setting for stochastic calculus on manifold we show that the convergence can be realized as a strong one in the path space. Hopefully such a more precise convergence to Brownian motion will help us in understanding asymptotic behavior of some classical functional inequalities for path spaces.


Mar. 22
Jian Ding (UPenn)
Random walk among Bernoulli obstacles

Consider a discrete time simple random walk on \(Z^d, d\geq 2\) with random Bernoulli obstacles, where the random walk will be killed when it hits an obstacle. We show that the following holds for a typical environment (for which the origin is in an infinite cluster free of obstacles): conditioned on survival up to time n, the random walk will be localized in a single island. In addition, the limiting shape of the island is a ball and the asymptotic volume is also determined. This is based on joint works with Changji Xu. Time permitting, I will also describe a recent result in the annealed case, which is a joint work with Ryoki Fukushima, Rongfeng Sun and Changji Xu.


Mar. 29
Tai Melcher (UVA)
Convergence rates for the empirical spectral distribution of Brownian motion on the unitary group

In 1997 Biane showed that Brownian motion on the unitary group \(U(N)\) converges as a process to the "free unitary Brownian motion" as \(N\) gets large. A corollary of this result is the convergence of the empirical spectral distribution of a unitary Brownian motion to a deterministic probability measure which can be described as the spectral measure of a free unitary Brownian motion. We will discuss recent results bounding the rates of convergence of these measures for large \(N\) and fixed time, and also as measure-valued paths. This is joint work with Elizabeth Meckes.


Apr. 5
Nayeong Kong (Lehigh)
Spectral Distribution of Random Graphs

In this talk, random geometric graphs on the unit sphere are considered. The adjacency matrix of these graphs can be viewed as a kind of random inner product matrix. Given this, we are able to use a result of Cheng and Singer to prove the following theorem. [Kong 2018] For the adjacency matrix \(A_n\) of the geometric graph on the unit sphere, we have $$ m_{A_n}(z)\rightarrow m(z) = \frac{-z+\sqrt{z^2-4c}}{2c}, \quad n\rightarrow\infty .$$ Here \(c\) is a constant appearing in the definition of the ensemble. What this means is that the ESD of the adjacency matrices of the given random geometric graphs converges to the famous Wigner semicircle distribution.


Apr. 19
Chongliang Luo (UPenn)
Applications of likelihood based methods in EHR and genetic data analysis

In this talk I will introduce two examples of using likelihood based methods for estimation bias reduction and hypothesis testing, with applications in electronic health record (EHR) data and genetic data analyses. The first example regards the bias reduction of estimation when measurement error exists and a small validation sample is available. By utilizing the influence functions and asymptotic normality of three non-optimal estimators we can construct an augmented estimator which is unbiased and more efficient. The second example regards a non-standard hypothesis testing problem when the null hypothesis lies on the boundary of the parameter space. A likelihood ratio test is used but its distribution under null is no longer chi square. This test can be applied in geno-pheno association test. By incorporating some reasonable constraints on the effects, the test is expected to achieve greater power for discovering association between genotypes and phenotypes.


Apr. 26
Antonio Auffinger (Northwestern)
To be rescheduled for the fall

This page is maintained on behalf of the probability and statistics group by Robert Neel; please email robert.neel(at)lehigh.edu with any quesions.