Free Book: Applied Stochastic Processes

  • Construction of Time-Continuous Stochastic Processes
  • From Random Walks to Brownian Motion
  • Stationarity, Ergodicity, Fractal Behavior
  • Memory-less or Markov Property
  • Non-Brownian Process
  • Integrated, Moving Average and Differential Process
  • Proper Re-scaling and Variance Computation
  • Application to Number Theory Problem
  • Controlled or Constrained Random Walks
  • Link to Mixture Distributions and Clustering
  • First Glimpse of Stochastic Integral Equations
  • Link to Wiener Processes, Application to Fintech
  • Potential Areas for Research
  • Non-stochastic Case
  • Gap Distribution in Pseudo-Random Digits
  • Statistical Testing and Geometric Distribution
  • Algorithm to Compute Gaps
  • Another Application to Number Theory Problem
  • Counter-Example: Failing the Gap Test
  • Graph Theory and Network Processes
  • The Six Degrees of Separation Problem
  • Programming Languages Failing to Produce Randomness in Simulations
  • How to Identify and Fix the Previous Issue
  • Application to Web Crawling
  • Logistic Map and Fractals
  • Simulation: Flaws in Popular Random Number Generators
  • Quantum Algorithms
  • General Framework
  • Equilibrium Distribution and Stochastic Integral Equation
  • Examples of Chaotic Sequences
  • Discrete, Continuous Sequences and Generalizations
  • Special Logistic Map
  • Auto-regressive Time Series
  • Literature
  • Source Code with Big Number Library
  • Solving the Stochastic Integral Equation: Example
  • Precision Issues when Simulating, Modeling, and Analyzing Chaotic Processes
  • When Precision Matters, and when it does not
  • High Precision Computing (HPC)
  • Benchmarking HPC Solutions
  • How to Assess the Accuracy of your Simulation Tool
  • Application: Random Number Generation
  • Chaotic Sequences Representing Numbers
  • Data Science and Mathematical Engineering
  • Numbers in Base 2, 10, 3/2 or p
  • Nested Square Roots and Logistic Map
  • About the Randomness of the Digits of p
  • The Digits of p are Randomly Distributed in the Logistic Map System
  • Paths to Proving Randomness in the Decimal System
  • Connection with Brownian Motions
  • Randomness and the Bad Seeds Paradox
  • Application to Cryptography, Financial Markets, Blockchain, and HPC
  • Digits of p in Base p
  • Summary Table: Equilibrium Distribution, Properties
  • Reverse-engineering Number Representation Systems
  • Application to Cryptography
  • Components of Number Representation Systems
  • General Properties of these Systems
  • Examples of Number Representation Systems
  • Examples of Patterns in Digits Distribution
  • Defects found in the Logistic Map System
  • Test of Uniformity
  • New Numeration System with no Bad Seed
  • Holes, Autocorrelations, and Entropy (Information Theory)
  • Towards a more General, Better, Hybrid System
  • Faulty Digits, Ergodicity, and High Precision Computing
  • Finding the Equilibrium Distribution with the Percentile Test
  • Central Limit Theorem, Random Walks, Brownian Motions, Stock Market Modeling
  • Data Set and Excel Computations
  • A Special Case of the Central Limit Theorem
  • Simulations, Testing, and Conclusions
  • Generalizations
  • Source Code
  • Central Limit Theorem for Non-Random Variables
  • Testing Randomness: Max Gap, Auto-Correlations and More
  • Potential Research Areas
  • Generalization to Higher Dimensions
  • Simulations
  • Theoretical Distribution of Records over Time
  • How and Why: Decorrelate Time Series
  • A Weird Stochastic-Like, Chaotic Sequence
  • Stochastic Geometry, Spatial Processes, Random Circles: Coverage Problem
  • Additional Reading (Including Twin Points in Point Processes)

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Vincent Granville

Vincent Granville

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Founder, MLtechniques.com. Machine learning scientist. Co-founder of Data Science Central (acquired by Tech Target).