# MSCS Seminars Today

## Calendar for Wednesday November 22, 2017

Wednesday November 22, 2017

**Statistics Seminar**

The tail asymptotics of the Brownian signature

Xi Geng (Carnegie Mellon University)

3:00 PM in SEO 636

In the groundbreaking work of B. Hambly and T. Lyons (Uniqueness for the signature of a path of bounded variation and the reduced path group, Ann. of Math., 2010), it has been conjectured that the geometry of a tree-reduced bounded variation path can be recovered from the tail asymptotics of its associated sequence of iterated path integrals. While this conjecture is still remaining open in the general deterministic case, in this talk we investigate a similar problem in the probabilistic setting for Brownian motion. It turns out that a martingale approach applied to the hyperbolic development of Brownian motion allows us to extract useful information from the tail asymptotics of Brownian iterated integrals, which can be used to determined the Brownian rough path along with its natural parametrization uniquely. This in particular strengthens the existing uniqueness results in the literature.

**Complex Analysis Seminar**

The 1-dimensional extension property in complex analysis

Mark Lawrence (Nazarbayev University)

3:00 PM in SEO 1227

A classical theorem states that if a function on the unit circle has
vanishing negative Fourier coefficients, then it extends to holomorphic
function on the unit disc. What happens when you are given a family of
curves, and a function which extends holomorphically from each of the
curves? This area of study is called the "1-dimensional extension
problem". Results for planar domains, and for holomorphic extension from
boundaries in C^n will be discussed. One application is the construction
of a completely new class of algebras of real analytic functions.
Various techniques of analytic extension in several variables are used
to prove these results.

**Statistics Seminar**

Causality in the joint analysis of longitudinal and survival data

Lei Liu (Washington University in St. Louis )

4:00 PM in SEO 636

In many biomedical studies, disease progress is monitored by a biomarker over time, e.g., repeated measures of CD4, hemoglobin level in end stage renal disease (ESRD) patients. The endpoint of interest, e.g., death or diagnosis of a specific disease, is correlated with the longitudinal biomarker. The causal relation between the longitudinal and time to event data is of interest. In this paper we examine the causality in the analysis of longitudinal and survival data. We consider four questions: (1) whether the longitudinal biomarker is a mediator between treatment and survival outcome; (2) whether the biomarker is a surrogate marker; (3) whether the relation between biomarker and survival outcome is purely due to an unknown confounder; (4) whether there is a mediator moderator for treatment. We illustrate our methods by data from two clinical trials: an AIDS study and a liver cirrhosis study.