Weird Random Walks: Synthetizing, Testing and Leveraging Quasi-randomness

Entitled “Weird Random Walks: Synthetizing, Testing and Leveraging Quasi-randomness”, the full version in PDF format is accessible in the “Free Books and Articles” section, here.

I discuss different types of synthetized random walks that are almost perfectly random, in one and two dimensions. Besides the theoretical interest, it provides new modeling tools, especially for physicists, engineers, natural sciences, security, fintech and quant professionals.

The kind of irregularities injected in these random walks are especially weak and hard to detect. The research results presented here are new, focused on applications, and state-of-the-art. In addition to offering original modeling tools, these unusual stochastic processes can be used to benchmark fraud detection systems or to benchmark tests of randomness.

The picture below features a metric that magnifies the very weak patterns, to show that despite all appearances, something is “off”, and definitely not random in my simulated random walks. You can fine-tune various parameters in the accompanying Python code, to produce different types of non-randomness, ranging from totally undetectable to hard to detect.

Delta metric for 4 types of random walks: perfect randomness would correspond to delta = 0!


This is a follow-up to my article “Detecting Subtle Departures from Randomness”, where I introduced the prime test to identify very weak violations of various laws of large numbers. Pseudo-random sequences failing this test usually pass most test batteries, yet are unsuitable for a number of applications, such as security, strong cryptography, or intensive simulations. The purpose here is to build such sequences with very low, slow-building, long-range dependencies, but that otherwise appear as random as pure noise. They are useful not only for testing and benchmarking tests of randomness, but also in their own right to model almost random systems, such as stock market prices. I introduce new categories of random walks (or quasi-Brownian motions subject to constraints), and discuss the peculiarities of each category. For completeness, I included related stochastic processes discussed in some of my previous articles, for instance integrated and 2D clustered Brownian motions. All the processes investigated here are drift-free and symmetric, yet not perfectly random. They all start at zero.

Table of Contents

Symmetric unbiased constrained random walks

  • Three fundamental properties of pure random walks
  • Random walks with more entropy than pure random signal
  • Applications
  • Algorithm to generate quasi-random sequences
  • Variance of the modified random walk
  • Random walks with less entropy than pure random signal

Related stochastic processes

  • From Brownian motions to clustered Lévy flights
  • Integrated Brownian motions and special autoregressive processes

Python code

  • Computing probabilities and variances
  • Path simulations

Download the Article

The technical article, entitled Weird Random Walks: Synthetizing, Testing and Leveraging Quasi-randomness, is accessible in the “Free Books and Articles” section, here. The text highlighted in orange in this PDF document are keywords that will be incorporated in the index, when I aggregate all my related articles into a single book about innovative machine learning techniques. The text highlighted in blue corresponds to external clickable links, mostly references. And red is used for internal links, pointing to a section, bibliography entry, equation, and so on.

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About the Author

Vincent Granville is a pioneering data scientist and machine learning expert, co-founder of Data Science Central (acquired by  TechTarget in 2020), former VC-funded executive, author and patent owner. Vincent’s past corporate experience includes Visa, Wells Fargo, eBay, NBC, Microsoft, CNET, InfoSpace. Vincent is also a former post-doc at Cambridge University, and the National Institute of Statistical Sciences (NISS).  

Vincent published in Journal of Number TheoryJournal of the Royal Statistical Society (Series B), and IEEE Transactions on Pattern Analysis and Machine Intelligence. He is also the author of multiple books, available here. He lives  in Washington state, and enjoys doing research on stochastic processes, dynamical systems, experimental math and probabilistic number theory.

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