Central limit theorems under special relativity

Abstract

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 Wehn, Stroock and Varadhan, but in this special case the asymptotic distribution has an explicit form that is readily seen to exhibit lognormal tail behavior.

Central Limit Theorems under additive deformations

“…We have shown that different algebraic deformations with distinct isomorphisms yield different limiting distributions. The three statistical systems considered, the combination of velocity mentioned in [9], and α -norms of random variables all arise as special cases of our extended central limit theorems.…”

“…Our result can be used to derive a relativistic CLT given by [9], who considered the case of Kaniadakis addition: $$x\stackrel{\kappa }{\oplus }y=x\sqrt{1+{\kappa }^2{y}^2}+y\sqrt{1+{\kappa }^2{x}^2},$$representing the addition of momenta in special relativity. The parameter 0 < κ ≤ 1 is the reciprocal of the speed of light in the ambient space (when all variables are expressed in dimensionless units).…”

“…The parameter 0 < κ ≤ 1 is the reciprocal of the speed of light in the ambient space (when all variables are expressed in dimensionless units). The Lie group $\left(ℝ,\stackrel{\kappa }{\oplus }\right)$ has exponential map f κ ( x ) = sinh( κx )/ κ , and logarithmic map $$g_{\kappa }\left(x\right)=\frac{1}{\kappa }{\text{sinh}}^{-1}\left(\kappa x\right)=\frac{1}{\kappa }\text{log}(\kappa x+\sqrt{1+{\kappa }^2{x}^2}).$$Condition (3) can be checked using an inequality in [9], which yields $$\left|g_{\kappa }\left(x\right)-x\right|\le x^2\rho \left(x\right)$$with ρ ( x ) = κ min( κ | x |, 1) for all x ∈ ℝ. This ρ is bounded everywhere and ρ ( x ) → 0 as x → 0, so condition (2) holds.…”

“…[9] also derived CLTs for velocity and energy using identities of [7] and the continuous mapping theorem to translate the corresponding CLT for momentum into velocity and energy. Our Theorem 1 provides a more direct approach.…”

“…Tsallis entropy applies to statistical systems exhibiting the features of long range dependence [2], and has been successfully applied, for example, in image thresholding [3], modeling debris flow [4], analyzing electromagnetic pre-seismic emissions [5], and modeling the distribution of momenta of cold atoms in optical lattices [6]. Kaniadakis entropy arises when combining momenta in special relativity [7, 8], and its associated central limit theory has recently been developed by [9], who showed that the limiting distributions take the form of hyperbolic functions of standard normals.…”