Distributions of the total signal strengths from 1, 10, 100, and 10⁵ sources whose independent signal strengths have identical tails (left). Distribution of 𝒫_circ for 1, 10 and 100 identical sources (right). The dashed line shows the distribution from a single source, the solid lines show the analytic estimates, and the distributions obtained from an explicit Monte Carlo analysis are shown by the points.