The talk will focus on the standard exponential families of probability measures on Euclidean spaces that have finite supports. In such a family parameterized by means, the mean is supposed to move along a segment inside the convex support towards an endpoint on the boundary of the support. Limit behavior of several quantities related to the exponential family can be described explicitly, in particular, the variance functions and information divergences.

Shannon's mutual information between two random variables is a fundamental and venerable concept in information and communication theory, statistics and beyond. What is a measure of mutual dependence among an arbitrary number of random variables? A notion of ``shared information'' among mutiple terminals that observe correlated random variables and communicate interactively among themselves, is shown to play a useful role in certain problems of distributed processing and computation. A larger role for shared information, which for two terminals particularizes to mutual information, is an open question. This talk is based on joint works with Imre Csiszar, Sirin Nitinawarat, Himanshu Tyagi and Shun Watanabe.

Modeling and analysis of clustering recurrent event data are of great importance in seismology, sociology, medical science, finance and other areas, such as the modeling and predictions of earthquakes and other disastrous events, study of the patterns of neural firings in neuroscience, assessing the efficacy of cancer medications in suppressing the recurrence of tumors, predicting message propagation in Twitter, and analysis of the risk of default on debt repayments by borrowers. One important tool in statistical modeling of such event clusters is the Howkes process, or more generally, self-exciting process. This talk presents a brief review of some of the models and their analysis and applications.