Large-system consistency conjecture for the effective signal-count estimator

Let ${\bf R}$ be an $n\times n$ covariance matrix satisfying the hypothesis of the paper's spiked-convergence proposition, and let $\widehat{{\bf R}}$ be a sample covariance matrix formed from $m$ snapshots. For $c>0$, define the effective number of identifiable signals by

$$k_{\mathrm{eff}}(c\mid {\bf R}):= \text{the number of eigenvalues of }{\bf R}\text{ greater than }\lambda(1+\sqrt{c}).$$

Let $c_m=n/m$ and suppose that $c_m\to c$ as $m,n\to\infty$. Write $\widehat{k}$ for the estimate of the number of signals obtained using the proposed algorithm.

Large-system consistency conjecture. In this joint limit, $\widehat{k}$ is a consistent estimator of $k_{\mathrm{eff}}(c\mid {\bf R})$.

The claim concerns identifiability in the large-system, large-sample regime. The authors state that they cannot prove consistency because a refined analysis of fluctuations of subsets of ordered noise eigenvalues is needed, while numerical simulations provide only non-definitive corroborating evidence.