Sabir Umarov

Tufts University

A joint event between the New Hampshire Chapter of the IEEE Communications and Signal Processing and the Boston Chapter of the American Statistical Association.

Dinner: Appetizers and Cash Bar - Martha's Exchange, Nashua, NH  5:30 pm

Lecture: 6:30 pm

Talk Location:  Nashua Public Library

Abstract: 

The classical central limit theorem (CLT) and  /alpha/-stable 

distributions have rich applications in theoretical probabilistic and 

statistical studies, as well as various applied sciences including 

statistical mechanics, biostatistics, signal processing, econometrics, 

etc. As is known, the  density function of a normal distribution with a 

given mean and variance maximizes the Boltzmann-Gibbs-Shannon (BGS) 

entropy.  The Gaussian distribution is thus the maxent solution for BGS 

entropy. In 1988 an entropic form generalizing the BGS entropy was 

introduced by physicist C. Tsallis. The Tsallis' entropic form depends 

on a real parameter /q/, recovering the BGS entropy in the limit case 

/q/=1. The maxent for Tsallis' entropy for a given /q/ under some 

natural constraints is a /q/-Gaussians (deformed exponential), with 

either compact-support or heavy-tail decay.  These distributions arise 

in complex systems, impacted sometimes by nonlinear processes, and play 

an important role in physical descriptions of market trading, biological 

processes, and correlated noise sources. 

In the talk we discuss /q/-Gaussians and their properties, several 

versions of /q/-central limit theorems, which generalize the classical 

central limit theorem, and/ q-alpha/-stable distributions. Applications 

of this novel theory have been increasing over the last decade. Some of 

these applications will be presented in the talk.