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On the upper limits for dipole anisotropies in cosmic-ray positrons

The excess of cosmic-ray positrons in the energy range from 10 GeV to few hundred GeV reported by PAMELA and AMS experiments is not consistent with a pure secondary origin and requires the introduction of a source term. The presence of anisotropies in the positron arrival directions would be a distinctive signature of their origin. Current measurements are consistent with isotropy and limits to a dipole anisotropy have been established. In this note, we review the mathematical basis of this analysis and provide a general bound to the dipole upper limits achievable from a given sample of events. The published experimental limits are confronted with this bound.