More Weird Statistical Distributions

Some original and very interesting material is presented here, with possible applications in Fintech. No need for a PhD in math to understand this article: I tried to make the presentation as simple as possible, focusing on high-level results rather than technicalities. Yet, professional statisticians and mathematicians, even academic researchers, will find some deep and fascinating results worth further exploring.

Can you identify patterns in this chart? (see section 2.2. in the article for an answer)

Let’s start with

Here the X(k)’s are random variable identically and independently distributed, commonly referred to as X. We are trying to find the distribution of Z.

Contents

1. Using a Simple Discrete Distribution for X

2. Towards a Better Model

• Approximate Solution
• The Fractal, Brownian-like Error Term

3. Finding X and Z Using Characteristic Functions

• Test with Log-normal Distribution for X
• Playing with the Characteristic Functions
• Generalization to Continued Fractions and Nested Cubic Roots

4. Exercises

Read this article here.