Probability Theory and Stochastic Processes

A Full Year Graduate Course

Todd Kemp

Recorded Lectures

The Lectures for this course are pre-recorded, and available on YouTube.

The videos are organized by topic, concept, or example. They are each approximately 20-30 minutes in length. They are labeled according to a schedule of 2 traditional lectures per week in each of three 10-week quarter courses. Each lecture number is broken into 2 or 3 pieces; hence Lectures 30.1 and 30.2, or Lectures 57.1, 57.2, and 57.3.

Below, you will find a list (with links) of the lecture videos in chronological order, along with pdf slides of the tablet output during those lecture videos.

Lecture VideosSlides (Before)Slides (After)
Banach Tarski Banach Tarski 0. (Before) 0. (After)
Probability Motivation 1.1 (Before) 1.1 (After)
$\sigma$-Fields 1.2 (Before) 1.2 (After)
Measures: Definition and Examples 2.1 (Before) 2.1 (After)
Finitely Additive Measures 2.2 (Before) 2.2 (After)
Stieltjes Premeasures 3.1 (Before) 3.1 (After)
Outer Measure 3.2 (Before) 3.2 (After)
Outer Pseudo-Metric 4.1 (Before) 4.1 (After)
The Extension Theorem 4.2 (Before) 4.2 (After)
Uniqueness, and $\sigma$-Finite Extension 4.3 (Before) 4.3 (After)
Radon Measures 5.1 (Before) 5.1 (After)
Lebesgue Measure 5.2 (Before) 5.2 (After)
Random Variables Motivation 6.1 (Before) 6.1 (After)
Measurable Functions 6.2 (Before) 6.2. (After)
Robustness of Measurability 7.1 (Before) 7.1 (After)
Riemann-Stieltjes Integration 7.2 (Before) 7.2 (After)
Simple Integration 8.1 (Before) 8.1 (After)
The Lebesgue Integral and the MCTheorem 8.2 (Before) 8.2 (After)
Fatou and Borel-Cantelli 9.1 (Before) 9.1 (After)
$L^1$ and the Dominated Convergence Theorem 9.2 (Before) 9.2 (After)
Integrals and Derivatives 10.1 (Before) 10.1 (After)
Lebesgue vs. Riemann 10.2 (Before) 10.2 (After)
Radon-Nikodym 11.1 (Before) 11.1 (After)
Laws Revisited 11.2 (Before) 11.2 (After)
$L^2$ and Covariance 12.1 (Before) 12.1 (After)
The Weak Law of Large Numbers 12.2 (Before) 12.2 (After)
Convergence in Measure 13.1 (Before) 13.1 (After)
$L^p$ is Complete 13.2 (Before) 13.2 (After)
Dynkin's Multiplicative Systems Theorem 14.1 (Before) 14.1 (After)
Product Measure 14.2 (Before) 14.2 (After)
Tonelli's and Fubini's Theorems 14.3 (Before) 14.3 (After)
Independence 15.1 (Before) 15.1 (After)
Independent Random Variables 15.2 (Before) 15.2 (After)
Kolmogorov's Extension Theorem (I) 16.1 (Before) 16.1 (After)
Kolmogorov's Extension Theorem (II) 16.2 (Before) 16.2 (After)
Kolmogorov's 0-1 Law 17.1 (Before) 17.1 (After)
Convolution 17.2 (Before) 17.2 (After)
Strong Law of Large Numbers (I) 18.1 (Before) 18.1 (After)
Kolmogorov's Convergence Criterion 18.2 (Before) 18.2 (After)
Strong Law of Large Numbers (II) 19.1 (Before) 19.1 (After)
Renewal Theory 19.2 (Before) 19.2 (After)
Cramer's LDP (I) 20.1 (Before) 20.1 (After)
Cramer's LDP (II) 20.2 (Before) 20.2 (After)
Total Variation 21.1 (Before) 21.1 (After)
Law of Rare Events 21.2 (Before) 21.2 (After)
Weak Convergence 22.1 (Before) 22.1 (After)
Weak Convergence over $\mathbb{R}^d$ 22.2 (Before) 22.2 (After)
Vague Convergence 23.1 (Before) 23.1 (After)
Prokhorov's Compactness Theorem 23.2 (Before) 23.2 (After)
Skorohod's Theorem 23.3 (Before) 23.3 (After)
Complex Integration and Dynkin's Theorem 24.1 (Before) 24.1 (After)
Characteristic Function 24.2 (Before) 24.2 (After)
The Riemann-Lebesgue Lemma 25.1 (Before) 25.1 (After)
Fourier Inversion 25.2 (Before) 25.2 (After)
The Continuity Theorem 25.3 (Before) 25.3 (After)
The Central Limit Theorem 26.1 (Before) 26.1 (After)
Infinite Divisibility 26.2 (Before) 26.2 (After)
Average Uniformity 27.1 (Before) 27.1 (After)
Lindberg-Feller CLT 27.2 (Before) 27.2 (After)
Random Permutations 27.3 (Before) 27.3 (After)
Metrics on Probability Measures 28.1 (Before) 28.1 (After)
Stein's Method 28.2 (Before) 28.2 (After)
Berry-Esseen Theorem 29.1 (Before) 29.1 (After)
Local Dependencies 29.2 (Before) 29.2 (After)
Conditional Probability 30.1 (Before) 30.1 (After)
Conditional Expectation (I) 30.2 (Before) 30.2 (After)
Orthogonal Projection 31.1 (Before) 31.1 (After)
Conditional Expectation (II) 31.2 (Before) 31.2 (After)
Conditioning as Partial Integration 32.1 (Before) 32.1 (After)
Conditional MCT, Fatou, DCT, Jensen 32.2 (Before) 32.2 (After)
Conditioning on Random Variables 32.3 (Before) 32.3 (After)
Probability Kernels (I) 33.1 (Before) 33.1 (After)
Regular Conditional Distributions 33.2 (Before) 33.2 (After)
Probability Kernels (II) 34.1 (Before) 34.1 (After)
Random Dynamics 34.2 (Before) 34.2 (After)
Stochastic Processes 35.1 (Before) 35.1 (After)
Poisson Process 35.2 (Before) 35.2 (After)
The Markov Property 36.1 (Before) 36.1 (After)
Probability Kernels Revisited 36.2 (Before) 36.2 (After)
Markov Processes 37.1 (Before) 37.1 (After)
Kolmogorov's (Extended) Extension Theorem 37.2 (Before) 37.2 (After)
Path Space 38.1 (Before) 38.1 (After)
Time Homogeneous Processes 38.2 (Before) 38.2 (After)
Markov Matrix 39.1 (Before) 39.1 (After)
Markov Generator 39.2 (Before) 39.2 (After)
Operator Norm Continuity 40.1 (Before) 40.1 (After)
Bounded Generators 40.2 (Before) 40.2 (After)
Generator Matrix 41.1 (Before) 41.1 (After)
Compound Poisson Process 41.2 (Before) 41.2 (After)
Jump-Hold Description 42.1 (Before) 42.1 (After)
Hitting Times 42.2 (Before) 42.2 (After)
First Step Recursion 43.1 (Before) 43.1 (After)
Biased Random Walks 43.2 (Before) 43.2 (After)
Finite Expectation Hitting Times 44.1 (Before) 44.1 (After)
Invariant Distributions 44.2 (Before) 44.2 (After)
Stopping Times 45.1 (Before) 45.1 (After)
Stopped Processes 45.2 (Before) 45.2 (After)
Stopping Times and Conditioning 45.3 (Before) 45.3 (After)
The Strong Markov Property 46.1 (Before) 46.1 (After)
Markov Chain Ergodic Theorem 46.2 (Before) 46.2 (After)
Wald's Identity 46.3 (Before) 46.3 (After)
Betting Strategies 47.1 (Before) 47.1 (After)
Martingales 47.2 (Before) 47.2 (After)
Uniform Integrability 48.1 (Before) 48.1 (After)
Markov Chains and Martingales 48.2 (Before) 48.2 (After)
Optional Stopping and Sampling 49.1 (Before) 49.1 (After)
Holder's Inequality 49.2 (Before) 49.2 (After)
Submartingale Maximal Inequalities 49.3 (Before) 49.3 (After)
Doob's Upcrossing Inequality 50.1 (Before) 50.1 (After)
Martingale Convergence Theorem 50.2 (Before) 50.2 (After)
Versions 51.1 (Before) 51.1 (After)
Holder Continuity 51.2 (Before) 51.2 (After)
Kolmogorov Continuity Criteria 51.3 (Before) 51.3 (After)
Wiener Measure 52.1 (Before) 52.1 (After)
Kolmogorov Tightness Criteria 52.2 (Before) 52.2 (After)
Weak Convergence on Path Space 52.3 (Before) 52.3 (After)
Weak Convergence Odds and Ends 53.1 (Before) 53.1 (After)
Random Walk CLT 53.2 (Before) 53.2 (After)
Donsker's Functional CLT 53.3 (Before) 53.3 (After)
$p$-Variation 54.1 (Before) 54.1 (After)
Rough Paths 54.2 (Before) 54.2 (After)
Gaussian Processes 55.1 (Before) 55.1 (After)
Scaling Brownian Motion 55.2 (Before) 55.2 (After)
Progressively Measurable Processes 56.1 (Before) 56.1 (After)
Stopping and Optional times 56.2 (Before) 56.2 (After)
Stopped Processes 56.3 (Before) 56.3 (After)
Strong Markov Property (I) 57.1 (Before) 57.1 (After)
Strong Markov Property (II) 57.2 (Before) 57.2 (After)
Strong Markov Property (III) 57.3 (Before) 57.3 (After)
Blumenthal's 0-1 Law 58.1 (Before) 58.1 (After)
Brownian Paths Oscillate Rapidly 58.2 (Before) 58.2 (After)
The Reflection Principle 58.3 (Before) 58.3 (After)
Bachelier's Principle and the Arcsin Law 59.1 (Before) 59.1 (After)
The Dirichlet Problem 59.2 (Before) 59.2 (After)
Where Do We Go From Here? 60. (Before) 60. (After)