link to University of Nebraska Lincoln
Department of Mathematics
203 Avery Hall
University of Nebraska Lincoln
Lincoln, NE 68588-0323
402-472-3731 (voice)
402-472-8466 (fax)

Topics in
Probability Theory and Stochastic Processes
Steven R. Dunbar


Lessons

Note: In all notes, lessons, tests, quizzes and homework assignments, unless specifically noted otherwise

  1. The lessons use MathML, use a modern browser to view.
  2. The mathematical function log(x) means the natural logarithm, that is, logarithms to the base e.

Real Analysis Background

  1. Asymptotic Limits and Series
    Updated Monday, 03-Oct-2011 06:02:34 CDT
  2. PDF Version: Asymptotic Limits and Series
    Updated Monday, 03-Oct-2011 06:02:27 CDT

Stirling's Formula

  1. A Survey of Stirling's Formula
    Note: PDF file
    Updated Monday, 23-Apr-2012 05:32:53 CDT
  2. Wallis Formula (HTML with Mathjax)
    Updated Wednesday, 18-Nov-2015 09:20:24 CST
  3. Wallis Formula (PDF) Wallis Formula
    Updated Wednesday, 18-Nov-2015 09:20:32 CST
  4. Evaluation of the Gaussian Density Integral
    Updated Saturday, 22-Oct-2011 05:49:19 CDT
  5. PDF Version: Evaluation of the Gaussian Density Integral
    Updated Saturday, 22-Oct-2011 05:49:13 CDT
  6. (MathJax Version) Stirling's Formula Derived from Elementary Sequences and Series
    Updated Monday, 06-Jan-2014 06:07:25 CST
  7. PDF Version: Stirling's Formula Derived from Elementary Sequences and Series
    Updated Monday, 06-Jan-2014 06:10:27 CST
  8. Stirling's Formula by Using Euler-Maclaurin Summation
    Updated Monday, 10-Oct-2011 06:00:47 CDT
  9. PDF Version: Stirling's Formula by Using Euler-Maclaurin Summation
    Updated Monday, 10-Oct-2011 06:00:40 CDT
  10. Stirling's Formula from Wallis' Formula and the Trapezoidal Approximation
    Updated Monday, 10-Oct-2011 06:00:00 CDT
  11. PDF Version: Stirling's Formula from Wallis' Formula and the Trapezoidal Approximation
    Updated Monday, 10-Oct-2011 05:59:56 CDT
  12. Stirling's Formula from Sums and Taylor Series Approximation
    Updated Friday, 16-Dec-2011 07:39:51 CST
  13. PDF Version: Stirling's Formula from Sums and Taylor Series Approximation
    Updated Friday, 16-Dec-2011 07:39:44 CST
  14. Stirling's Formula Derived from the Gamma Function
    Updated Monday, 26-Nov-2012 05:56:59 CST
    PDF Version: Stirling's Formula Derived from the Gamma Function
    Updated Monday, 26-Nov-2012 05:57:13 CST
  15. Stirling's Formula in Real and Complex Variables
    Updated Monday, 23-May-2011 05:55:15 CDT
    PDF Version: Stirling's Formula in Real and Complex Variables
    Updated Monday, 23-May-2011 05:55:25 CDT
  16. Stirling's Formula Derived from the Poisson Distribution
    Updated [an error occurred while processing this directive]
    PDF Version: Stirling's Formula Derived from the Poisson Distribution
    Updated Friday, 23-Jun-2017 06:01:05 CDT

Bernoulli Trials

  1. Binomial Distribution
    Updated Monday, 30-Jan-2017 09:05:15 CST
  2. PDF Version: Binomial Distribution
    Updated Monday, 30-Jan-2017 09:05:17 CST
  3. Weak Law of Large Numbers
    Updated Wednesday, 25-May-2011 06:16:28 CDT
  4. PDF Version: Weak Law of Large Numbers
    Updated Wednesday, 25-May-2011 06:16:34 CDT
  5. Large Deviations
    Updated Saturday, 04-Jun-2011 09:38:07 CDT
  6. PDF Version: Large Deviations
    Updated Saturday, 04-Jun-2011 09:38:15 CDT
  7. de Moivre Laplace Central Limit Theorem
    Updated Friday, 09-Dec-2011 21:16:09 CST
  8. PDF Version: de Moivre Laplace Central Limit Theorem
    Updated Friday, 09-Dec-2011 21:15:59 CST
  9. The Moderate Deviations Result
    Updated Thursday, 29-Nov-2012 05:51:24 CST
  10. PDF Version: The Moderate Deviations Result
    Updated Thursday, 29-Nov-2012 05:51:35 CST
  11. MathJax version The Local Limit Theorem
    Updated Sunday, 26-Apr-2015 09:32:17 CDT
  12. PDF Version: The Local Limit Theorem
    Updated Sunday, 26-Apr-2015 09:32:08 CDT
  13. PDF Version: The Ballot Theorem and the Reflection Principle
    Updated Tuesday, 31-Dec-2013 09:05:13 CST
  14. MathJax Version: The Ballot Theorem and the Reflection Principle
    Updated Friday, 11-Dec-2015 11:35:57 CST
  15. PDF Version: The Hitting Time Theorem
    Updated Tuesday, 15-Aug-2017 05:28:53 CDT
  16. MathJax Version: The Hitting Time Theorem
    Updated Tuesday, 15-Aug-2017 05:28:53 CDT
  17. PDF Version: Positive Walks
    Updated Wednesday, 24-Sep-2014 07:37:26 CDT
  18. MathJax Version: Positive Walks
    Updated Wednesday, 24-Sep-2014 07:39:53 CDT
  19. PDF Version: Arcsine Law
    Updated Tuesday, 10-Mar-2015 05:52:23 CDT
  20. MathJax Version: Arcsine Law
    Updated Tuesday, 10-Mar-2015 05:52:38 CDT
  21. PDF Version: Almost Sure Events
    Updated Friday, 12-Jun-2015 07:45:28 CDT
  22. MathJax Version: Almost Sure Events
    Updated Friday, 12-Jun-2015 07:45:34 CDT
  23. PDF Version: Strong Law of Large Numbers
    Updated Wednesday, 30-Dec-2015 08:20:45 CST
  24. MathJax Version: Strong Law of Large Numbers
    Updated Wednesday, 30-Dec-2015 08:20:45 CST

Normal Random Variables, Gaussian Distributions

  1. The Sum of Independent Normal Random Variables is Normal
  2. Updated Saturday, 24-Jun-2017 09:15:41 CDT

Brownian Motion

  1. Introduction to Bernstein Polynomials and Brownian Motion
  2. Bernstein Polynomials and the Weierstrass Approximation Theorem
  3. Bernstein Motions and the convergence to Wiener measure

Central Limit Theorem

  • PDF Version: The Sum of Independent Normal Random Variables is Normal
  • Updated Saturday, 24-Jun-2017 09:15:41 CDT

    Brownian Motion

    1. PDF Version: Introduction to Bernstein Polynomials and Brownian Motion
    2. PDF Version: Bernstein Polynomials and the Weierstrass Approximation Theorem
    3. PDF Version: Bernstein Motions and the convergence to Wiener measure

    Central Limit Theorem

    Large Deviations

    Markov Process Variants

    PDF Version: Review of Markov Processes
    Updated Monday, 27-Mar-2017 11:04:06 CDT

    PDF Version: Waiting Time to Absorption
    Updated Monday, 30-Jan-2017 05:59:17 CST

    Eigenvalues, Eigenvectors, and Normal Forms of Matrices
    PDF Version: Eigenvalues, Eigenvectors, and Normal Forms of Matrices
    Updated Friday, 28-May-2010 05:47:02 CDT


    PDF Version:

    Fastest Mixing Markov Chain
    Updated Friday, 16-Apr-2010 06:03:37 CDT
    PDF Version: Fastest Mixing Markov Chain
    Updated Friday, 16-Apr-2010 06:03:31 CDT


    Hidden Markov Models

    Examples of Hidden Markov Models
    Updated Wednesday, 19-Apr-2017 07:45:22 CDT
    PDF Version: Examples of Hidden Markov Models
    Updated Friday, 14-Apr-2017 08:18:38 CDT

    PDF : Talk/Slides about Examples of Hidden Markov Models
    Updated Friday, 14-Apr-2017 08:35:33 CDT

    Notation and Problems of Hidden Markov Models
    Updated Wednesday, 19-Apr-2017 07:46:02 CDT
    PDF Version: Examples of Hidden Markov Models
    Updated Friday, 14-Apr-2017 08:19:05 CDT

    PDF : Talk/Slides about Notation and Problems of Hidden Markov Models
    Updated Friday, 14-Apr-2017 08:35:36 CDT

    Algorithms for Hidden Markov Models
    Updated Tuesday, 09-May-2017 09:15:15 CDT
    PDF Version: Algorithms for Hidden Markov Models
    Updated Tuesday, 09-May-2017 09:15:11 CDT

    PDF : Talk/Slides about Algorithms for Hidden Markov Models
    Updated Friday, 03-Mar-2017 09:24:45 CST

    Applications, especially with Markov Chains

    Worst Case and Average Case Behavior of the Simplex Algorithm
    Updated Thursday, 20-Jan-2011 21:01:31 CST
    PDF Version: Worst Case and Average Case Behavior of the Simplex Algorithm
    Updated Thursday, 20-Jan-2011 21:01:27 CST

    Smoothed Analysis for Linear Optimization Algorithm
    Updated Thursday, 27-Jan-2011 20:52:43 CST
    PDF Version: Smoothed Analysis for Linear Optimization Algorithm
    Updated Thursday, 27-Jan-2011 20:52:41 CST

    Are the Digits in Mersenne Primes Random? A Probability Model.
    Updated Wednesday, 12-Oct-2016 08:36:11 CDT

    Martingales

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    Weak Law of Large Numbers

    Counting and Poisson Processes

    Conditional Probability and Expectation

    Online Tables for Probability and Cumulative Distributions

    Here are some links to Online Distribution Tables:
    1. Statsoft This site has several different distributions in a nice back-of-the-book-looking HTML format.
    2. David Lane, Rice University, Hyperstat This links to a Java applet that will compute the percentages for the normal distribution when given the Z values

    Annotated Bibliography

    A personal, eclectic and biased annotated bibliography of books and articles about probability, stochastic processes and mathematical finance.

    Outside Readings/Links:

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    [ Steve Dunbar's home page ] Steve Dunbar's Home Page, http://www.math.unl.edu/~sdunbar
    [ Email to Steve Dunbar ] Email to Steve Dunbar, sdunbar@unl.edu