Limiting distribution under equicorrelation covariance structures

Characterize the limiting distribution of the relevant high-dimensional covariance statistic when the covariance matrix has the equicorrelation form $(1-\rho)I_d+\rho 1_d1_d^\top$ with $\rho\in(0,1)$ and the fourth-trace condition used in prior work fails.

Background

In discussing related covariance-testing results, the paper considers covariance structures of the form $(1-\rho)I_d+\rho 1_d1_d^\top$. For this structure, the ratio $$\operatorname{tr}(\Gamma_n^4)/\operatorname{tr}^2(\Gamma_n^2)$$ does not vanish, so a condition imposed in earlier work is unavailable. The paper explicitly notes that the limiting distribution in this setting is unresolved; the present paper provides Gaussian approximation results for its own U-statistic norm framework but does not characterize that previously unclear limiting distribution.