An elementary renormalization-group approach to the generalized central limit theorem and extreme value distributions
Ariel Amir
Abstract
Abstract The generalized central limit theorem is a remarkable generalization of the central limit theorem, showing that the sum of a large number of independent, identically-distributed (i.i.d) random variables with infinite variance may converge under appropriate scaling to a distribution belonging to a special family known as Lévy stable distributions. Similarly, the maximum of i.i.d. variables may converge to a distribution belonging to one of three universality classes (Gumbel, Weibull and Fréchet). Here, we rederive these known results following a mathematically non-rigorous yet highly transparent renormalization-group-inspired approach that captures both of these universal results following a nearly identical procedure.
Topics & Concepts
Central limit theoremMathematicsExtreme value theoryRandom variableUniversality (dynamical systems)GeneralizationLimit (mathematics)ScalingIllustration of the central limit theoremWeibull distributionGeneralized extreme value distributionApplied mathematicsStatistical physicsPure mathematicsDistribution (mathematics)Scaling limitExchangeable random variablesProbability distributionVariance (accounting)Mathematical analysisDiscrete mathematicsConvergence of random variablesDonsker's theoremCombinatoricsEmpirical distribution functionNormal distributionRandom Matrices and ApplicationsFinancial Risk and Volatility ModelingProbability and Risk Models