A Randomized Central Limit Theorem

Iddo Eliazar*, Joseph Klafter

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


The Central Limit Theorem (CLT), one of the most elemental pillars of Probability Theory and Statistical Physics, asserts that: the universal probability law of large aggregates of independent and identically distributed random summands with zero mean and finite variance, scaled by the square root of the aggregate-size (sqrt(n)), is Gaussian. The scaling scheme of the CLT is deterministic and uniform - scaling all aggregate-summands by the common and deterministic factor sqrt(n). This Letter considers scaling schemes which are stochastic and non-uniform, and presents a "Randomized Central Limit Theorem" (RCLT): we establish a class of random scaling schemes which yields universal probability laws of large aggregates of independent and identically distributed random summands. The RCLT universal probability laws, in turn, are the one-sided and the symmetric Lévy laws.

Original languageEnglish
Pages (from-to)290-293
Number of pages4
JournalChemical Physics
Issue number1-3
StatePublished - 12 May 2010


  • Central Limit Theorem (CLT)
  • One-sided Lévy laws
  • Poisson processes
  • Power-laws
  • Randomized Central Limit Theorem (RCLT)
  • Symmetric Lévy laws
  • Universality


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