The Theory of Probability | December 2025

true

The Theory of Probability

Santosh S. Venkatesh

5 Sep 2025 to 22 Nov 2025

About the Book

From its classical foundations to the advanced modern theory, this self-contained and comprehensive guide to probability weaves together mathematical proofs, historical context, and richly detailed illustrative applications. It is suitable for advanced undergraduates with a solid mathematics preparation, graduate students in theoretical disciplines, and researchers in pure and applied probability who wish to stay abreast of the remarkable developments of the late twentieth century.

The book uniquely combines rigorous mathematical treatment with engaging historical anecdotes and concrete examples that bring probability theory to life. Topics range from basic probability spaces to advanced concepts like algorithmic randomness and Brownian motion.

You can find this book at Santosh Venkatesh’s website.

Prerequisites

A foundation in mathematical analysis and linear algebra is recommended. The book appears to assume some familiarity with advanced calculus and basic concepts from real analysis.

Timings and Venue

We start with an introduction session on Friday, September 5th, followed by regular sessions Monday through Friday at 9AM IST for about 60-90 minutes each day according to the schedule below.

I am very tentatively setting aside a week per chapter, and this will get us through Part A over ten weeks. The last week is reserved for review, although we’ll probably take those as buffer days for missed sessions.

I plan to use this page to maintain a tentative schedule. I hope to post rough discussion summaries. For announcements and access to session recordings, please join us on Circle.

Note: unlike previous reading groups, there are no Zoom links: the online sessions happen through Circle. Also, no Whatsapp groups! Circle will send you email reminders ahead of the sessions and you can also add the events to your calendar.

See you there!

Date	Lecture	Slides	Notes	Video
05 Sep, 2025	Introduction Session Welcome and overview of the book and reading group format
09 Sep, 2025	Probability Spaces (Part 1) Introduction to probability spaces and basic concepts
10 Sep, 2025	Probability Spaces (Part 2) Sigma-algebras and measurable spaces
11 Sep, 2025	Probability Spaces (Part 3) Probability measures and their properties
12 Sep, 2025	Probability Spaces (Part 4) Random variables and distributions
13 Sep, 2025	Probability Spaces (Part 5) Expectation and moments
16 Sep, 2025	Conditional Probability (Part 1) Basic concepts of conditional probability
17 Sep, 2025	Conditional Probability (Part 2) Bayes' theorem and its applications
18 Sep, 2025	Conditional Probability (Part 3) Conditional expectation
19 Sep, 2025	Conditional Probability (Part 4) Martingales and stopping times
20 Sep, 2025	Conditional Probability (Part 5) Filtrations and information
23 Sep, 2025	A First Look at Independence (Part 1) Introduction to independence
24 Sep, 2025	A First Look at Independence (Part 2) Product measures and independent events
25 Sep, 2025	A First Look at Independence (Part 3) Independent random variables
26 Sep, 2025	A First Look at Independence (Part 4) Sums of independent random variables
27 Sep, 2025	A First Look at Independence (Part 5) The law of large numbers
30 Sep, 2025	Probability Sieves (Part 1) Introduction to sieve methods
01 Oct, 2025	Probability Sieves (Part 2) Inclusion-exclusion principle
02 Oct, 2025	Probability Sieves (Part 3) Erdős–Ko–Rado theorem
03 Oct, 2025	Probability Sieves (Part 4) Sieve of Eratosthenes probabilistic version
04 Oct, 2025	Probability Sieves (Part 5) Large sieve inequality
07 Oct, 2025	Numbers Play a Game of Chance (Part 1) Probability in number theory
08 Oct, 2025	Numbers Play a Game of Chance (Part 2) Distribution of prime numbers
09 Oct, 2025	Numbers Play a Game of Chance (Part 3) Random matrix theory connections
10 Oct, 2025	Numbers Play a Game of Chance (Part 4) Probabilistic number theory results
11 Oct, 2025	Numbers Play a Game of Chance (Part 5) The circle method
14 Oct, 2025	The Normal Law (Part 1) Introduction to the normal distribution
15 Oct, 2025	The Normal Law (Part 2) Properties of the normal distribution
16 Oct, 2025	The Normal Law (Part 3) The central limit theorem
17 Oct, 2025	The Normal Law (Part 4) Proofs of the CLT
18 Oct, 2025	The Normal Law (Part 5) Local CLT and Berry-Esseen theorem
21 Oct, 2025	Probabilities on the Real Line (Part 1) Distribution functions and densities
22 Oct, 2025	Probabilities on the Real Line (Part 2) Lebesgue-Stieltjes measures
23 Oct, 2025	Probabilities on the Real Line (Part 3) Convolutions and characteristic functions
24 Oct, 2025	Probabilities on the Real Line (Part 4) Fourier analysis in probability
25 Oct, 2025	Probabilities on the Real Line (Part 5) Stable distributions
28 Oct, 2025	The Bernoulli Schema (Part 1) Bernoulli trials and binomial distribution
29 Oct, 2025	The Bernoulli Schema (Part 2) Poisson paradigm and rare events
30 Oct, 2025	The Bernoulli Schema (Part 3) Compound Poisson distributions
31 Oct, 2025	The Bernoulli Schema (Part 4) Branching processes
01 Nov, 2025	The Bernoulli Schema (Part 5) Galton-Watson process
04 Nov, 2025	The Essence of Randomness (Part 1) Concepts of randomness
05 Nov, 2025	The Essence of Randomness (Part 2) Kolmogorov complexity
06 Nov, 2025	The Essence of Randomness (Part 3) Algorithmic randomness
07 Nov, 2025	The Essence of Randomness (Part 4) Martin-Pour-El theory
08 Nov, 2025	The Essence of Randomness (Part 5) Chaitin’s Ω number
11 Nov, 2025	The Coda of the Normal (Part 1) Advanced properties of the normal distribution
12 Nov, 2025	The Coda of the Normal (Part 2) Multivariate normal distribution
13 Nov, 2025	The Coda of the Normal (Part 3) Gaussian processes
14 Nov, 2025	The Coda of the Normal (Part 4) Brownian motion
15 Nov, 2025	The Coda of the Normal (Part 5) Wiener process properties
18 Nov, 2025	Review Session 1 Review of Probability Spaces and Conditional Probability
19 Nov, 2025	Review Session 2 Review of Independence and Sieves
20 Nov, 2025	Review Session 3 Review of Number Theory and Normal Law
21 Nov, 2025	Review Session 4 Review of Real Line Probabilities and Bernoulli Schema
22 Nov, 2025	Review Session 5 Review of Randomness and Normal Coda