Higher-order statistics of the stochastic gravitational wave background from supermassive black hole binaries

Recent progress in gravitational wave observations has positioned pulsar timing arrays as a key tool for detecting the stochastic gravitational wave background in the nanohertz band. It is widely believed that this background is primarily attributed to the cosmic ensemble of inspiraling supermassive black hole binaries. While traditional analyses have predominantly focused on the spectral amplitude and frequency dependence of the gravitational wave background, higher-order statistics such as variance, skewness, and excess kurtosis could potentially be useful for extracting further physical information. However, these statistical moments are known to diverge when the redshift integration is extended down to $z=0$. In this study, we propose a strategy to resolve this issue by introducing a physically motivated lower integration limit, ${z}_{\mathrm{min}}$, defined by the sensitivity for detecting individual sources. Since higher-order statistics are primarily determined by local sources,