Probability Theory

Read these first

1. 01Characteristic Functions

   A complete re-encoding of a distribution as a function of frequency. It turns the messy business of adding independent random variables into plain multiplication, always exists even when the moment generating function does not, and is the machinery behind the central limit theorem.

2. 02Lebesgue Measure and Integration

   A way of measuring the size of sets and adding things up by sorting them into piles of equal value rather than sweeping left to right. It is the integral that makes expectation work for any random variable, mixes jumps and densities in one formula, and survives limits that break the Riemann integral.

3. 03Modes of Convergence for Random Variables

   Four different meanings of "the estimator settles down", almost sure, in probability, in mean square, and in distribution. They are not interchangeable, they sit in a strict hierarchy, and picking the wrong one is how proofs and simulations quietly go wrong.

4. 04Sigma-Algebras and Filtrations

   A sigma-algebra is the list of questions your information can answer; a filtration is that list growing as time passes. Together they are how probability theory writes down "what a trader knows at 10:30 this morning", the machinery behind martingales, conditional expectation and every no-look-ahead rule in backtesting.

5. 05The Radon-Nikodym Derivative

   The exchange rate between two probability measures, a per-scenario weight that converts one set of beliefs into another. It is what makes risk-neutral pricing, Girsanov's theorem, importance sampling and likelihood ratios all the same operation.

Then the rest

1. 06Kolmogorov's Axioms of Probability
2. 07Convolution and Sums of Random Variables
3. 08Independence and Product Measures
4. 09The Law of Total Variance
5. 10The Probability Integral Transform
6. 11Transformations of Random Variables
7. 12Change of Measure
8. 13Chernoff Bounds
9. 14Convergence in Distribution
10. 15Coupling and Total Variation Distance
11. 16Cumulants and the Cumulant Generating Function
12. 17Hoeffding's Inequality
13. 18Large Deviations and Cramér's Theorem
14. 19Monotone and Dominated Convergence Theorems
15. 20Poisson Approximation and the Chen-Stein Method
16. 21Probability Generating Functions
17. 22Stochastic Dominance
18. 23The Borel-Cantelli Lemmas
19. 24The Fubini-Tonelli Theorem
20. 25The Lindeberg-Feller Central Limit Theorem
21. 26The Wasserstein Distance