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Related Concept Videos

Absolute and Local Extreme Values01:22

Absolute and Local Extreme Values

The highest and lowest values of a function, relative to a reference axis, are known as extreme values. These include absolute maximum and absolute minimum values, which represent the highest and lowest points the function reaches across its entire domain. Within a restricted portion of the function, the highest and lowest values are referred to as local maximum and local minimum values, respectively.Periodic functions, such as sine and cosine, show extreme values at infinitely many points due...
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Resonance Raman Spectroscopy of Extreme Nanowires and Other 1D Systems
07:44

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Published on: April 28, 2016

Extreme-value distributions and renormalization group.

Iván Calvo1, Juan C Cuchí, J G Esteve

  • 1Laboratorio Nacional de Fusión, Asociación EURATOM-CIEMAT, 28040 Madrid, Spain. ivan.calvo@ciemat.es

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 11, 2012
PubMed
Summary
This summary is machine-generated.

This study explores generalized normalizations for extreme value theory, moving beyond traditional affine methods. It introduces renormalization-group transformations to analyze limit distributions and their properties.

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Area of Science:

  • Probability Theory
  • Statistical Mechanics
  • Extreme Value Theory

Background:

  • Classical extreme value theory focuses on affine normalization of random variable sequences.
  • Existing literature predominantly uses affine normalization, limiting broader applicability.
  • A need exists for more general normalization techniques in extreme value analysis.

Purpose of the Study:

  • To investigate and develop more general normalization methods for extreme value theory.
  • To apply renormalization-group transformations to analyze limit distributions.
  • To explore the mathematical and physical relevance of non-affine normalizations.

Main Methods:

  • Utilizing the framework of renormalization-group transformations in the space of probability densities.
  • Identifying limit distributions as fixed points of these transformations.
  • Analyzing the differential of the transformation for local insights.

Main Results:

  • Demonstrated the naturalness and utility of general normalizations beyond affine methods.
  • Established limit distributions as fixed points of renormalization-group transformations.
  • Enabled local analysis of domains of attraction and computation of finite-size corrections.

Conclusions:

  • Generalized normalizations offer a more comprehensive approach to extreme value theory.
  • Renormalization-group methods provide powerful tools for analyzing limit distributions.
  • This work extends the theoretical and practical scope of extreme value analysis.