Title:
Skew Products Over The Irrational Rotation, The Central Limit Theorem And RATs
Abstract:
Let f be a step function on the circle with zero mean and rational discontinuities while alpha is a quadratic irrational. The point-wise ergodic theorem tells us that the ergodic sums, f(x)+f(x+alpha)+...+f(x+(n-1)alpha) is o(n) for almost every x but says nothing about its deviations from zero, that is, its discrepancy; the study of these deviations naturally draws us to the study of ergodic transformations on infinite measure spaces, viz., skew products over irrational rotations. In this talk, after a brief introduction to these terms, we will learn how the temporal statistics of the ergodic sums for x=0 can be studied via random affine transformations (RATs) leading to a central limit theorem and other fine properties like the visit times to a neighbourhood of 0 vis-à-vis bounded rational ergodicity (all of course time permitting). This is reporting on joint work with Jon Aaronson and Michael Bromberg.
Time:
11:00am-12:00pm
Location:
Ramanujan Hall
Description:
Title: On a Poset of Trees I and II by Peter Csikvari
Abstract: We observe periodic phenomena everyday in our lives. The daily temperature of Delhi or the number of tourists visiting the famous Taj Mahal or the ECG data of a normal human being, clearly follow periodic nature. Sometimes, the observations may not be exactly periodic
due to different reasons, but they may be nearly periodic. The received data is usually disturbed by various factors. Due to random nature of the data, statistical techniques play important roles in analyzing the data. Statistics is also used in the formulation of appropriate models to describe the behavior of the system, development of an appropriate technique for estimation of model parameters, and the assessment of model performances. In this talk we will discuss different techniques which we have developed for the last twenty five years for analyzing periodic data, other than the standard Fourier analysis.