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

Asymptotic Limits
and Series
Updated Monday, 03Oct2011 06:02:34 CDT
 PDF Version:
Asymptotic Limits
and Series
Updated Monday, 03Oct2011 06:02:27 CDT
Stirling's Formula

A Survey of Stirling's Formula
Note: PDF file
Updated Monday, 23Apr2012 05:32:53 CDT

Wallis Formula (HTML with Mathjax)
Updated Friday, 03Nov2017 05:24:16 CDT
 Wallis Formula (PDF)
Wallis Formula
Updated Thursday, 02Nov2017 08:12:13 CDT

Evaluation of the Gaussian Density Integral
Updated Saturday, 22Oct2011 05:49:19 CDT
 PDF Version:
Evaluation of the Gaussian Density Integral
Updated Saturday, 22Oct2011 05:49:13 CDT

(MathJax Version) Stirling's Formula Derived from Elementary Sequences
and Series
Updated Monday, 06Jan2014 06:07:25 CST
 PDF Version:
Stirling's Formula Derived from Elementary Sequences
and Series
Updated Monday, 06Jan2014 06:10:27 CST

Stirling's Formula by Using EulerMaclaurin Summation
Updated Monday, 10Oct2011 06:00:47 CDT
 PDF Version:
Stirling's Formula by Using EulerMaclaurin Summation
Updated Monday, 10Oct2011 06:00:40 CDT

Stirling's Formula from Wallis' Formula and the
Trapezoidal Approximation
Updated Monday, 10Oct2011 06:00:00 CDT
 PDF Version:
Stirling's Formula from Wallis' Formula and the
Trapezoidal Approximation
Updated Monday, 10Oct2011 05:59:56 CDT

Stirling's Formula from Sums and Taylor Series Approximation
Updated Friday, 16Dec2011 07:39:51 CST
 PDF Version:
Stirling's Formula from Sums and Taylor Series Approximation
Updated Friday, 16Dec2011 07:39:44 CST

Stirling's Formula Derived from the Gamma Function
Updated Monday, 26Nov2012 05:56:59 CST
PDF Version:
Stirling's Formula Derived from the Gamma Function
Updated Monday, 26Nov2012 05:57:13 CST

Stirling's Formula in Real and Complex Variables
Updated Monday, 23May2011 05:55:15 CDT
PDF Version:
Stirling's Formula in Real and Complex Variables
Updated Monday, 23May2011 05:55:25 CDT

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, 23Jun2017 06:01:05 CDT
Bernoulli Trials

Binomial Distribution
Updated Tuesday, 11Sep2018 05:34:57 CDT
 PDF Version:
Binomial Distribution
Updated Tuesday, 11Sep2018 05:35:03 CDT

Weak Law of Large Numbers
Updated Wednesday, 25May2011 06:16:28 CDT
 PDF Version:
Weak Law of Large Numbers
Updated Wednesday, 25May2011 06:16:34 CDT

Large Deviations
Updated Friday, 09Nov2018 05:45:27 CST
 PDF Version:
Large Deviations
Updated Friday, 09Nov2018 05:45:31 CST

de Moivre Laplace Central Limit Theorem
Updated Friday, 09Dec2011 21:16:09 CST
 PDF Version:
de Moivre Laplace Central Limit Theorem
Updated Friday, 09Dec2011 21:15:59 CST

The Moderate Deviations Result
Updated Thursday, 29Nov2012 05:51:24 CST
 PDF
Version:
The Moderate Deviations Result
Updated Thursday, 29Nov2012 05:51:35 CST
 MathJax version
The Local Limit Theorem
Updated Sunday, 26Apr2015 09:32:17 CDT
 PDF
Version:
The Local Limit Theorem
Updated Sunday, 26Apr2015 09:32:08 CDT
 PDF
Version:
Ruin Probabilities
Updated Thursday, 28Sep2017 08:40:58 CDT
 MathJax
Version:
Ruin Probabilities
Updated Thursday, 28Sep2017 08:40:58 CDT
 PDF
Version:
Duration of Game
Updated Thursday, 28Sep2017 08:41:18 CDT
 MathJax
Version:
Duration of Game
Updated Thursday, 28Sep2017 08:41:18 CDT
 PDF
Version:
The Ballot Theorem and the Reflection Principle
Updated Tuesday, 31Dec2013 09:05:13 CST
 MathJax
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The Ballot Theorem and the Reflection Principle
Updated Friday, 11Dec2015 11:35:57 CST
 PDF
Version:
The Hitting Time Theorem
Updated Tuesday, 15Aug2017 05:28:53 CDT
 MathJax
Version:
The Hitting Time Theorem
Updated Tuesday, 15Aug2017 05:28:53 CDT
 PDF
Version:
Positive Walks
Updated Wednesday, 24Sep2014 07:37:26 CDT
 MathJax
Version:
Positive Walks
Updated Wednesday, 24Sep2014 07:39:53 CDT
 PDF
Version:
Arcsine Law
Updated Tuesday, 10Mar2015 05:52:23 CDT
 MathJax
Version:
Arcsine Law
Updated Tuesday, 10Mar2015 05:52:38 CDT
 PDF
Version:
Almost Sure Events
Updated Monday, 25Sep2017 13:41:30 CDT
 MathJax
Version:
Almost Sure Events
Updated Monday, 25Sep2017 13:41:34 CDT
 PDF
Version:
Strong Law of Large Numbers
Updated Friday, 20Oct2017 05:53:50 CDT
 MathJax
Version:
Strong Law of Large Numbers
Updated Friday, 20Oct2017 05:53:55 CDT
 PDF
Version:
BorelCantelli Lemmas with Examples
Updated Tuesday, 10Oct2017 08:23:32 CDT
 MathJax
Version:
BorelCantelli Lemmas with Examples
Updated Tuesday, 10Oct2017 08:23:38 CDT
 PDF
Version:
Law of the Iterated Logarithm
Updated Thursday, 22Mar2018 14:06:20 CDT
 MathJax
Version:
Law of the Iterated Logarithm
Updated Thursday, 22Mar2018 14:06:09 CDT
 PDF
Version:
Recurrence
Updated Wednesday, 21Feb2018 05:45:44 CST
 MathJax
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Recurrence
Updated Wednesday, 21Feb2018 05:45:41 CST
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Version:
Recurrence in Higher Dimensions
Updated Tuesday, 20Mar2018 12:08:59 CDT
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Version:
Recurrence in Higher Dimensions
Updated Tuesday, 20Mar2018 12:08:57 CDT
 PDF
Version:
Analytic Model of Coin Flipping
Updated Friday, 15Jun2018 07:49:04 CDT
 MathJax
Version:
Analytic Model of Coin Flipping
Updated Monday, 18Jun2018 07:34:03 CDT
 PDF
Version:
Analytic Proof of Central Limit Theorem
Updated Friday, 12Oct2018 05:47:11 CDT
 MathJax
Version:
Analytic Proof of Central Limit Theorem
Updated Friday, 12Oct2018 05:47:20 CDT
 PDF
Version:
Random Signs in Series, Random Series
Updated Tuesday, 26Jun2018 07:49:56 CDT
 MathJax
Version:
Random Signs in Series, Random Series
Updated Tuesday, 26Jun2018 07:58:50 CDT
Normal Random Variables, Gaussian Distributions

The Sum of Independent Normal Random Variables is Normal
Updated Saturday, 24Jun2017 09:15:41 CDT
Central Limit Theorem
PDF Version:
The Sum of Independent Normal Random Variables is Normal
Updated Saturday, 24Jun2017 09:15:41 CDT
Brownian Motion
 PDF Version:
Introduction
to Bernstein Polynomials and Brownian Motion
 PDF Version:
Bernstein Polynomials and the Weierstrass Approximation Theorem
 PDF Version:
Bernstein Motions and the convergence to Wiener measure
Stochastic Calculus
 PDF Version:
Examples of Stochastic Differential Equations
Central Limit Theorem
Large Deviations
Markov Chains
PDF Version:
Review of Markov Processes
Updated Monday, 27Mar2017 11:04:06 CDT
Classes of states and Stationary Distributions
PDF Version:
Classes of States and Stationary Distributions
Updated Monday, 25Sep2017 13:25:58 CDT
PDF Version:
Waiting Time to Absorption
Updated Monday, 30Jan2017 05:59:17 CST
Eigenvalues, Eigenvectors, and Normal Forms of Matrices
PDF Version:
Eigenvalues, Eigenvectors, and Normal Forms of Matrices
Updated Friday, 28May2010 05:47:02 CDT
PDF Version:
Fastest Mixing Markov Chain
Updated Friday, 16Apr2010 06:03:37 CDT
PDF Version:
Fastest Mixing Markov Chain
Updated Friday, 16Apr2010 06:03:31 CDT
Hidden Markov Models
Examples of Hidden Markov Models
Updated Wednesday, 19Apr2017 07:45:22 CDT
PDF
Version:
Examples of Hidden Markov Models
Updated Friday, 14Apr2017 08:18:38 CDT
PDF
:
Talk/Slides about Examples of Hidden Markov Models
Updated Friday, 14Apr2017 08:35:33 CDT
Notation and Problems of Hidden Markov Models
Updated Wednesday, 19Apr2017 07:46:02 CDT
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Version:
Examples of Hidden Markov Models
Updated Friday, 14Apr2017 08:19:05 CDT
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:
Talk/Slides about Notation and Problems of Hidden Markov Models
Updated Friday, 14Apr2017 08:35:36 CDT
Algorithms for Hidden Markov Models
Updated Tuesday, 09May2017 09:15:15 CDT
PDF
Version:
Algorithms for Hidden Markov Models
Updated Tuesday, 09May2017 09:15:11 CDT
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:
Talk/Slides about Algorithms for Hidden Markov Models
Updated Friday, 03Mar2017 09:24:45 CST
Applications, especially with Markov Chains
Distinguishing a Biased Coin from a Fair Coin
Updated Friday, 07Sep2018 08:00:19 CDT
PDF Version:
Distinguishing a Biased Coin from a Fair Coin
Updated Friday, 07Sep2018 08:00:13 CDT
Worst Case and Average Case Behavior of the Simplex
Algorithm
Updated Thursday, 20Jan2011 21:01:31 CST
PDF Version: Worst Case and Average Case Behavior of the Simplex
Algorithm
Updated Thursday, 20Jan2011 21:01:27 CST
Smoothed Analysis for Linear Optimization
Algorithm
Updated Thursday, 27Jan2011 20:52:43 CST
PDF Version:
Smoothed Analysis for Linear Optimization
Algorithm
Updated Thursday, 27Jan2011 20:52:41 CST
Are the Digits in Mersenne Primes Random? A Probability Model.
Updated Wednesday, 12Oct2016 08:36:11 CDT
Martingales
>
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:
 Statsoft
This site has several different distributions in a nice
backofthebooklooking HTML format.
 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, http://www.math.unl.edu/~sdunbar
Email to Steve Dunbar,
sdunbar@unl.edu