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.*