# 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
• 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

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## More from Vincent Granville

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

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## Vincent Granville

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