Asymptotic locations of bounded and unbounded eigenvalues of sample correlation matrices of certain factor models -- application to a components retention rule

Let the dimension N of data and the sample size T tend to ∞ with N/T → c > 0. The spectral properties of a sample correlation matrix C and a sample covariance matrix S are asymptotically equal whenever the population correlation matrix R is bounded (El Karoui 2009). We demonstrate this also for general linear models for unbounded R, by examining the behavior of the singular values of multiplicatively perturbed matrices. By this, we establish: Given a factor model of an idiosyncratic noise variance σ² and a rank-r factor loading matrix L which rows all have common Euclidean norm L. Then, the kth largest eigenvalues λₖ (1≤ k≤ N) of C satisfy almost surely: (1) λᵣ diverges, (2) λₖ/sₖ²→1/(L² + σ²) (1 ≤ k ≤ r) for the kth largest singular value sₖ of L, and (3) λr + 1→(1-ρ)(1+√c)² for ρ := L²/(L² + σ²). Whenever sᵣ is much larger than √log N, then broken-stick rule (Frontier 1976, Jackson 1993), which estimates rank\, L by a random partition (Holst 1980) of $[0,\,1]$, tends to r (a.s.). We also provide a natural factor model where the rule tends to "essential rank" of L (a.s.) which is smaller than rank\, L.