## Abstract

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 language | English |
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Pages (from-to) | 290-293 |

Number of pages | 4 |

Journal | Chemical Physics |

Volume | 370 |

Issue number | 1-3 |

DOIs | |

State | Published - 12 May 2010 |

## Keywords

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