Cosmic ray electrons and positrons from discrete stochastic sources

The distances that galactic cosmic ray electrons and positrons can travel are severely limited by energy losses to at most a few kiloparsec, thereby rendering the local spectrum very sensitive to the exact distribution of sources in our galactic neighbourhood. However, due to our ignorance of the exact source distribution, we can only predict the spectrum stochastically. We argue that even in the case of a large number of sources the central limit theorem is not applicable, but that the standard deviation for the flux from a random source is divergent due to a long power law tail of the probability density. Instead, we compute the expectation value and characterise the scatter around it by quantiles of the probability density using a generalised central limit theorem in a fully analytical way. The uncertainty band is asymmetric about the expectation value and can become quite large for TeV energies. In particular, the predicted local spectrum is marginally consistent with the measurements by Fermi-LAT and HESS even without imposing spectral breaks or cut-offs at source. We conclude that this uncertainty has to be properly accounted for when predicting electron fluxes above a few hundred GeV from astrophysical sources.