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Central limit theorems under special relativity. (English) Zbl 1323.60043

Summary: Several relativistic extensions of the Maxwell-Boltzmann distribution have been proposed, but they do not explain observed lognormal tail-behavior in the flux distribution of various astrophysical sources. Motivated by this question, extensions of classical central limit theorems are developed under the conditions of special relativity. The results are related to CLTs on locally compact Lie groups developed by D. Wehn [Proc. Natl. Acad. Sci. USA 48, 791–795 (1962; Zbl 0111.14104)] and D. W. Stroock and S. R. S. Varadhan [Sankhyā, Ser. A 35, 277–294 (1973; Zbl 0299.60007)], but in this special case the asymptotic distribution has an explicit form that is readily seen to exhibit lognormal tail behavior.

MSC:

60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
22E99 Lie groups
83A05 Special relativity
85A99 Astronomy and astrophysics
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