Thursday, March 21, 2019

Fascinating Developments in the Theory of Randomness

I present here some innovative results in my most recent research on stochastic processes. chaos modeling, and dynamical systems, with applications to Fintech, cryptography, number theory, and random number generators. While covering advanced topics, this article is accessible to professionals with limited knowledge in statistical or mathematical theory. It introduces new material not covered in my recent book (available here) on applied stochastic processes. You don't need to read my book to understand this article, but the book is a nice complement and introduction to the concepts discussed here.
None of the material presented here is covered in standard textbooks on stochastic processes or dynamical systems. In particular, it has nothing to do with the classical logistic map or Brownian motions, though the systems investigated here exhibit very similar behaviors and are related to the classical models. This cross-disciplinary article is targeted to professionals with interests in statistics, probability, mathematics, machine learning, simulations, signal processing, operations research, computer science, pattern recognition, and physics. Because of its tutorial style, it should also appeal to beginners learning about Markov processes, time series, and data science techniques in general, offering fresh, off-the-beaten-path content not found anywhere else, contrasting with the material covered again and again in countless, identical books, websites, and classes catering to students and researchers alike. 
Some problems discussed here could be used by college professors in the classroom, or as original exam questions, while others are extremely challenging questions that could be the subject of a PhD thesis or even well beyond that level. This article constitutes (along with my book) a stepping stone in my endeavor to solve one of the biggest mysteries in the universe: are the digits of mathematical constants such as Pi, evenly distributed? To this day, no one knows if these digits even have a distribution to start with, let alone whether that distribution is uniform or not. Part of the discussion is about statistical properties of numeration systems in a non-integer base (such as the golden ratio base) and its applications. All systems investigated here, whether deterministic or not, are treated as stochastic processes, including the digits in question. They all exhibit strong chaos, albeit easily manageable due to their ergodicity.  .
Interesting connections with the golden ratio, special polynomials, and other special mathematical constants, are discussed in section 2. Finally, all the analyses performed during this work were done in Excel. I share my spreadsheets in this article, as well as many illustration, and all the results are replicable.
Content of this article
1. General framework, notations and terminology
  • Finding the equilibrium distribution
  • Auto-correlation and spectral analysis
  • Ergodicity, convergence, and attractors
  • Space state, time state, and Markov chain approximations
  • Examples
2. Case study
  • First fundamental theorem
  • Second fundamental theorem
  • Convergence to equilibrium: illustration
3. Applications
  • Potential application domains
  • Example: the golden ratio process
  • Finding other useful b-processes
4. Additional research topics
  • Perfect stochastic processes
  • Characterization of equilibrium distributions (the attractors)
  • Probabilistic calculus and number theory, special integrals
5. Appendix
  • Computing the auto-correlation at equilibrium
  • Proof of the first fundamental theorem
  • How to find the exact equilibrium distribution
6. Additional Resources

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