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.
