Probability and Statistics
Foundations of probability, stochastic processes, and statistical inference.
Probability and Statistics. Foundations of probability, stochastic processes, and statistical inference. The literature on probability and statistics divides naturally along several axes: the foundational structures that organise the subject, the techniques that drive proofs and computations, the questions about classification or representation that animate current research, and the bridges to neighbouring areas of mathematics and science. The references below trace those axes through the canonical textbook treatments and recent technical contributions.
Foundations and canonical references
The standard treatments of probability and statistics 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. Ross, A First Course in Probability (2014) offers an alternative presentation that complements the primary references and is useful for triangulating definitions and proof techniques. Billingsley, Probability (1995) gives a parallel, more proof-oriented exposition of the same material and is widely used as a graduate text.
Open methodological questions for probability and statistics 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 · supporting · 2014A First Course in Probabilityross-2014
- textbook · primary · 1995Probabilitybillingsley-1995
In context
Where this topic sits in the prerequisite graph. Click any node to jump.
Explore
- 01
Probability Theory
Sigma-algebras, random variables, expectation, and limit theorems.
- 02
Stochastic Processes
Random processes indexed by time: Markov chains, Lévy, and Gaussian processes.
- 03
Stochastic Optimization
Optimisation of objective functions that depend on random variables — covering scenario-based stochastic programming, sample-path methods, distributionally robust formulations, and the stochastic-gradient family.
- 04
Stochastic Differential Equations
Differential equations driven by Brownian motion and other noise processes — covering existence and uniqueness theory, asymptotic and large-deviation analysis, statistical inference, and deep-learning solvers for high-dimensional and backward problems.
- 05
Mathematical Statistics
Estimation, hypothesis testing, and asymptotic theory.
- 06
Bayesian Statistics
Priors, posteriors, hierarchical models, and Bayesian computation.
- 07
High-Dimensional Statistics
Statistics when p ≫ n: sparsity, regularization, and minimax bounds.
- 08
Causal Inference
Potential outcomes, DAGs, and identification of causal effects.
- 09
Probabilistic Models on Combinatorial Structures
Statistical mechanics of disordered systems and random combinatorial structures.
- 10
Extreme Value Theory
Limit laws for maxima, tail estimation, and applications to risk.
- 11
Time Series Analysis
ARMA, state-space models, and nonstationary time series.
- 12
Experimental Design
Factorial, optimal, and sequential experimental designs.
- 13
Spatial Statistics
Geostatistics, point patterns, and lattice processes.
- 14
Functional Data Analysis
Statistics for curves and surfaces as data objects.
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