## Monday, September 10, 2018

### New Perspective on the Central Limit Theorem and Statistical Testing

You won't learn this in textbooks, college classes, or data camps. Some of the material in this article is very advanced yet presented in simple English, with an Excel implementation for various statistical tests, and no arcane theory, jargon, or obscure theorems. It has a number of applications, in finance in particular. This article covers several topics under a unified approach, so it was not easy to find a title. In particular, we discuss:
• When the central limit theorem fails: what to do, and case study
• Various original statistical tests, some unpublished, for instance to test if an empirical statistical distribution (based on observations) is symmetric or not, or whether two distributions are identical
• The power and mysteries of stable (also called divisible) statistical distributions
• Dealing with weighted sums of random variables, especially with decaying weights
• Fun number theory problems and algorithms associated with these statistical problems
• Decomposing a (theoretical or empirical / observed) statistical distribution into elementary components, just like decomposing a complex molecule into atoms
The focus is on principles, methodology, and techniques applicable to, and useful in many applications. For those willing to do a deeper dive on these topics, many references are provided. This article, written as a tutorial, is accessible to professionals with elementary statistical knowledge, like stats 101. It is also written in a compact style, so that you can grasp all the material in hours rather than days. This simple article covers topics that you could learn in MIT, Stanford, Berkeley, Princeton or Harvard classes aimed at PhD students. Some is state-of-the-art research results published here for the first time, and made accessible to the data science of data engineer novice. I think mathematicians (being one myself) will also enjoy it. Yet, emphasis is on applications rather than theory.
Finally, we focus here on sums of random variables.  The next article will focus on mixtures rather than sums, providing more flexibility for modeling purposes, or to decompose a complex distribution in elementary components. In both cases, my approach is mostly non-parametric, and based on robust statistical techniques, capable of handling outliers without problems, and not subject to over-fitting.
Content
1. Central Limit Theorem: New Approach
2. Stable and Attractor Distributions
• Using decaying weights
• More about stable distributions and their applications
3. Non CLT-compliant Weighted Sums, and their Attractors
• Testing for normality
• Testing for symmetry and dependence on kernel
• Testing for semi-stability
• Conclusions