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Graduate Seminar

Kevin Ou
Carnegie Mellon University
Title: Characterize Characteristic Functions of Probability Distributions

Abstract: Most probability texts give properties of characteristic functions (necessary conditions) but touch the converse (sufficient conditions) only tangentially. We will together re-discover and prove (rigorously) the most popular sufficient and necessary conditions of characteristic functions, called Bochner's Theorem (or Bochner-Khintchine in Russian texts). We then apply the theorem to two examples, one of which is the "real" Khintchine's Criterion. The related but harder moment problem may be mentioned (to motivate possible future talks).

Motivations to study Bochner's Theorem and/or to attend the talk: 1. Characteristic function plays a "central" role in probability theory. 2. Construct "possibly new" probability distributions. 3. Prepare to tackle the moment problem. 4. More accessible approach to an abstract Fourier analysis theorem (Fourier analysis will only be "$epsilon \downarrow 0$", and everyone in the department is welcomed) 5. Basic exams, maybe...

Date: Tuesday, September 22, 2015
Time: 5:30 pm
Location: Wean Hall 8220
Submitted by:  Zilin Jiang