Large deviation theory

Large deviation theory studies probabilities of rare events as a system’s size or observation time grows, identifying their exponential decay rates through rate functions.

Large deviation principle: An asymptotic statement that assigns exponentially scaled probabilities to sets through a rate function. It is the central mathematical form used to express large-deviation estimates.
Law of large numbers: A theorem stating that sample averages converge in probability to an expected value under suitable assumptions. It describes typical behavior, while large deviations quantify departures from that behavior.
Sanov's theorem: A large deviation theorem describing empirical distributions of independent samples through relative entropy. It quantifies how unlikely an empirical distribution far from the true law is.
Moderate deviations: An asymptotic regime between central-limit fluctuations and large deviations, with its own scaling and rate estimates. It bridges Gaussian-scale fluctuations and the rarer exponential-scale events treated here.
Herman Chernoff: An American statistician known for foundational work on exponential bounds and hypothesis testing. Chernoff’s exponential-bound methods helped establish tools central to rare-event estimates.
Rate function: A function that assigns nonnegative costs to outcomes, governing their exponential probability decay. Its values specify the exponential cost of each atypical outcome.
Central limit theorem: A theorem describing the limiting distribution of suitably normalized sums of independent random variables. It captures common fluctuations, unlike the exponential-scale tails studied here.
Donsker–Varadhan theory: A framework for large deviations of empirical measures and occupation times of Markov processes. It extends rare-event rate calculations to long-running stochastic dynamics.
Extreme value theory: The study of limiting distributions for maxima and minima of large collections of random variables. It focuses on extremes, while large deviation theory addresses exponential probabilities more broadly.
Frank Spitzer: An American mathematician whose work shaped probability theory, including random walks and Markov processes. His work helped develop probabilistic methods later central to large-deviation analysis.

Linked from 8 pages