A geometric theory for Levy distributions

作者：Eliazar Iddo^*

来源：Annals of Physics, 2014, 347: 261-286.

DOI：10.1016/j.aop.2014.05.005

• Levy distributions
• Law of Large Numbers (LLN)
• Central Limit Theorem (CLT)
• Poisson processes
• Scaling
• Universality

Levy distributions are of prime importance in the physical sciences, and their universal emergence is commonly explained by the Generalized Central Limit Theorem (CLT). However, the Generalized CLT is a geometry-less probabilistic result, whereas physical processes usually take place in an embedding space whose spatial geometry is often of substantial significance. In this paper we introduce a model of random effects in random environments which, on the one hand, retains the underlying probabilistic structure of the Generalized CLT and, on the other hand, adds a general and versatile underlying geometric structure. Based on this model we obtain geometry-based counterparts of the Generalized CLT, thus establishing a geometric theory for Levy distributions. The theory explains the universal emergence of Levy distributions in physical settings which are well beyond the realm of the Generalized CLT.

• 出版日期2014-8