Probability

Probability studies random phenomena through distributions, expectation, independence, concentration, and stochastic processes.

The subject starts from limit theorems for sums of independent variables. Laws of large numbers give almost-sure convergence, with the strong law and Kolmogorov's three-series theorem, and central limit theorems give convergence in law, with the Berry–Esseen theorem bounding the error by \(C\rho/(\sigma^3\sqrt{n})\). Markov chains include the convergence theorem for finite irreducible aperiodic chains, and stochastic calculus has the Itô integral and its isometry.

Percolation is the largest field, with about an eighth of the articles. For Bernoulli bond percolation on \(\mathbb{Z}^d\) with \(d \ge 2\) the critical probability lies strictly between \(0\) and \(1\) (phase transition); below it, the probability that the origin is connected to distance \(n\) decays exponentially in \(n\) (sharpness); and on \(\mathbb{Z}^2\) at \(p = 1/2\) the Russo–Seymour–Welsh theorem keeps the probability of crossing a rectangle of fixed aspect ratio away from \(0\) and \(1\). Correlation inequalities supply the Harris–FKG and van den Berg–Kesten inequalities and Reimer's inequality, and high-dimensional probability has bounds on the KLS constant of log-concave laws.

Distributions, the second largest field, works out the moments, mass functions and tails of named laws. About a sixth of the subject is statistics: hypothesis testing, linear models and multivariate analysis, statistical estimation, Bayesian statistics around the Dirichlet process, and nonparametric and survival methods such as the Kaplan–Meier estimator and the Cox proportional hazards model.

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