Fractal Poisson processes

Iddo Eliazar, Joseph Klafter

Research output: Contribution to journalArticlepeer-review

Abstract

The Central Limit Theorem (CLT) and Extreme Value Theory (EVT) study, respectively, the stochastic limit-laws of sums and maxima of sequences of independent and identically distributed (i.i.d.) random variables via an affine scaling scheme. In this research we study the stochastic limit-laws of populations of i.i.d. random variables via nonlinear scaling schemes. The stochastic population-limits obtained are fractal Poisson processes which are statistically self-similar with respect to the scaling scheme applied, and which are characterized by two elemental structures: (i) a universal power-law structure common to all limits, and independent of the scaling scheme applied; (ii) a specific structure contingent on the scaling scheme applied. The sum-projection and the maximum-projection of the population-limits obtained are generalizations of the classic CLT and EVT results - extending them from affine to general nonlinear scaling schemes.

Original languageEnglish
Pages (from-to)4985-4996
Number of pages12
JournalPhysica A: Statistical Mechanics and its Applications
Volume387
Issue number21
DOIs
StatePublished - 1 Sep 2008

Keywords

  • Central Limit Theorem (CLT)
  • Extreme Value Theory (EVT)
  • Fractal Poisson processes
  • Nonlinear scaling
  • Power-laws
  • Self-similarity
  • Stochastic limit-laws

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