A Law of Fractional Logarithm for Nested Complex Sample Covariance Matrices

Abstract

I prove an almost-sure law of fractional logarithm and the associated finite-tail cluster sets for the largest eigenvalue along a northwest-nested path of complex sample covariance matrices drawn from one infinite array of independent entries. The entries have zero mean, unit variance, vanishing complex second moment, and uniformly bounded moments of every fixed order, but are not assumed Gaussian; the row dimension follows a nondecreasing bounded-increment path with a positive limiting aspect ratio. After the finite-$N$ soft-edge centering and scaling, the upper and lower fluctuation endpoints are $(1/4)^{2/3}$ on the $(\log N)^{2/3}$ scale and $-4^{1/3}$ on the $(\log N)^{1/3}$ scale, and the corresponding finite-tail cluster sets are $[0,(1/4)^{2/3}]$ and $-4^{1/3},\infty)$. These numerical values and set geometry agree with the established $β=2$ Wigner-minor law. The covariance setting requires a separate argument because rows and columns grow simultaneously, the matrices share a revealed northwest past, and the rectangular linearization has several orientations, including wide cases with deterministic zero modes. I use Gaussian Laguerre moderate-deviation estimates and the companion paper's separated-grid moment result. I then transfer the resulting occurrence bounds through a conditional comparison that keeps the past fixed, connect the added row and column strips to the Gaussian reference family, and apply recurrence on a common filtration. A deterministic no-downward-jump argument promotes the endpoint limits to the full cluster sets.