All You Need is not $\Omega_\mathrm{gw}$: Beyond the Mean of the Cross-Correlation Estimator when Searching for an Astrophysical Gravitational-Wave Background
Searches for the stochastic gravitational-wave background (SGWB) using ground-based detectors rely heavily on the cross-correlation estimator $\bar C_f$, whose statistical properties determine the theoretical foundation for inference frameworks built on it. Past analyses usually assume Gaussianity of $\bar C_f$, but this assumption has not been systematically tested for a background of astrophysical origin like the one produced by compact binary coalescences. In this work, we decompose $\bar C_f$ into three physically motivated components: (1) the mean intensity, (2) the geometrical shot noise, and (3) the polarization leakage, and we compute 90\% credible intervals for its cumulants up to fourth order for the binary black hole, binary neutron star, and neutron star-black hole populations by propagating local merger rate uncertainties. We find that for the upcoming LIGO O5 sensitivity, $\bar C_f$ remains effectively Gaussian for all three populations, validating current Gaussian likelihood frameworks. For Cosmic Explorer (CE), however, the $\bar C_f$ produced by binary black holes becomes non-Gaussian, with skewness and excess kurtosis of $0.31^{+0.04}_{-0.03}$ and $1.3^{+0.4}_{-0.3}$, respectively, in the most sensitive band over a one-year observation period. For binary neutron star and neutron star - black hole mergers, $\bar C_f$ remains largely Gaussian even at CE. These results indicate that the Edgeworth expansion provides an adequate leading-order description of the non-Gaussianity expected in the next-generation era, while also motivating the development of more sophisticated statistical methods for a more accurate treatment.