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We show via both analytical calculation and numerical simulation that theoptimal cross-correlation statistic (OS) for stochasticgravitational-wave-background (GWB) searches using data from pulsar timingarrays follows a generalized chi-squared (GX2) distribution-i.e., a linearcombination of chi-squared distributions with coefficients given by theeigenvalues of the quadratic form defining the statistic. This observation isparticularly important for calculating the frequentist statistical significanceof a possible GWB detection, which depends on the exact form of thedistribution of the OS signal-to-noise ratio (S/N) $\hat\rho \equiv \hat A_{\rmgw}^2/\sigma_0$ in the absence of GW-induced cross correlations (i.e., the nulldistribution). Previous discussions of the OS have incorrectly assumed that theanalytic null distribution of $\hat\rho$ is well-approximated by a zero-meanunit-variance Gaussian distribution. Empirical calculations show that the nulldistribution of $\hat\rho$ has "tails" which differ significantly from thosefor a Gaussian distribution, but which follow (exactly) a GX2 distribution. So,a correct analytical assessment of the statistical significance of a potentialdetection requires the use of a GX2 distribution.

Analytic distribution of the optimal cross-correlation statistic for stochastic gravitational-wave-background searches using pulsar timing arrays