  We consider a $p$-dimensional, centered normal population such that all
variables have a positive variance $\sigma^2$ and any correlation coefficient
between different variables is a given nonnegative constant $\rho<1$. Suppose
that both the sample size $n$ and population dimension $p$ tend to infinity
with $p/n \to c>0$. We prove that the limiting spectral distribution of a
sample correlation matrix is Mar\v{c}enko-Pastur distribution of index $c$ and
scale parameter $1-\rho$. By the limiting spectral distributions, we rigorously
show the limiting behavior of widespread stopping rules Guttman-Kaiser
criterion and cumulative-percentage-of-variation rule in PCA and EFA. As a
result, we establish the following dichotomous behavior of Guttman-Kaiser
criterion when both $n$ and $p$ are large, but $p/n$ is small: (1) the
criterion retains a small number of variables for $\rho>0$, as suggested by
Kaiser, Humphreys, and Tucker [Kaiser, H. F. (1992). On Cliff's formula, the
Kaiser-Guttman rule and the number of factors. Percept. Mot. Ski. 74]; and (2)
the criterion retains $p/2$ variables for $\rho=0$, as in a simulation study
[Yeomans, K. A. and Golder, P. A. (1982). The Guttman-Kaiser criterion as a
predictor of the number of common factors. J. Royal Stat. Soc. Series D.
31(3)].
