  The superposition of many astrophysical gravitational wave (GW) signals below
typical detection thresholds baths detectors in a stochastic gravitational wave
background (SGWB). In this work, we present a Fourier space approach to compute
the frequency-domain distribution of stochastic gravitational wave backgrounds
produced by discrete sources. Expressions for the moment-generating function
and the distribution of observed (discrete) Fourier modes are provided. The
results are first applied to the signal originating from all the mergers of
compact stellar remnants (black holes and neutron stars) in the Universe, which
is found to exhibit a $-4$ power-law tail. This tail is verified in the
signal-to-noise ratio distribution of GWTC events. The extent to which the
subtraction of bright (loud) mergers gaussianizes the resulting confusion noise
of unresolved sources is then illustrated. The power-law asymptotic tail for
the unsubtracted signal, and an exponentially decaying tail in the case of the
SGWB, are also derived analytically. Our results generalize to any background
of gravitational waves emanating from discrete, individually coherent, sources.
