For large but finite source counts, the PDF of rescaled GWB characteristic strain squared follows the universal form N^{1/3} times the reflected map-Airy distribution evaluated at N^{1/3}(y-1), fully determined by the mean strain and a new cubic shot-noise scale.
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A practical theorem on gravitational-wave background statistics
For large but finite source counts, the PDF of rescaled GWB characteristic strain squared follows the universal form N^{1/3} times the reflected map-Airy distribution evaluated at N^{1/3}(y-1), fully determined by the mean strain and a new cubic shot-noise scale.