Probability Theory
Sigma-algebras, random variables, expectation, and limit theorems.
Probability Theory. Sigma-algebras, random variables, expectation, and limit theorems. This page collects canonical references that organise the subject and provide entry points to its main techniques.
Foundations and canonical references
The standard treatments of probability theory approach the subject from complementary angles. Durrett, Probability: Theory and Examples (2019) is the anchor reference for the subject and lays out the core definitions, theorems, and worked examples that practitioners return to. Fristedt, A Modern Approach to Probability Theory (1997) gives a parallel, more proof-oriented exposition of the same material and is widely used as a graduate text.
Open methodological questions for probability theory include sharpening the bridges between foundational theory and computational practice, extending classical results to broader or more structured settings, and integrating the techniques surveyed above with adjacent mathematical disciplines. The references listed in this page are the entry points that current work builds on.
Prerequisites
Sources
- textbook · primary · 2019Probability: Theory and Examplesdurrett-2019
- textbook · primary · 1997A Modern Approach to Probability Theoryfristedt-1997, gray-1997
In context
Where this topic sits in the prerequisite graph. Click any node to jump.
Explore
- 01
Measure-Theoretic Probability
Construction of probability measures, Kolmogorov extension, and conditional expectation.
- 02
Limit Theorems
Laws of large numbers, central limit theorems, and large deviations.
- 03
Concentration Inequalities
Chernoff, Hoeffding, Bernstein, McDiarmid, and Talagrand bounds.
- 04
Martingale Theory
Doob decomposition, optional stopping, and convergence theorems.
- 05
Large Deviation Theory
Cramér, Sanov, Varadhan's lemma, and rate functions.
- 06
Coupling Methods
Coupling for mixing, Stein's method, and exchangeable pairs.
- 07
Malliavin Calculus
Stochastic calculus of variations and applications to finance and SPDE.
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