  This paper studies the asymptotic spectral properties of the sample
covariance matrix for high dimensional compositional data, including the
limiting spectral distribution, the limit of extreme eigenvalues, and the
central limit theorem for linear spectral statistics. All asymptotic results
are derived under the high-dimensional regime where the data dimension
increases to infinity proportionally with the sample size. The findings reveal
that the limiting spectral distribution is the well-known Marchenko-Pastur law.
The largest (or smallest non-zero) eigenvalue converges almost surely to the
left (or right) endpoint of the limiting spectral distribution, respectively.
Moreover, the linear spectral statistics demonstrate a Gaussian limit.
Simulation experiments demonstrate the accuracy of theoretical results.
