  We estimate the global minimum variance (GMV) portfolio in the
high-dimensional case using results from random matrix theory. This approach
leads to a shrinkage-type estimator which is distribution-free and it is
optimal in the sense of minimizing the out-of-sample variance. Its asymptotic
properties are investigated assuming that the number of assets $p$ depends on
the sample size $n$ such that $\frac{p}{n}\rightarrow c\in (0,+\infty)$ as $n$
tends to infinity. The results are obtained under weak assumptions imposed on
the distribution of the asset returns, namely it is only required the fourth
moments existence. Furthermore, we make no assumption on the upper bound of the
spectrum of the covariance matrix. As a result, the theoretical findings are
also valid if the dependencies between the asset returns are described by a
factor model which appears to be very popular in financial literature nowadays.
This is also well-documented in a numerical study where the small- and
large-sample behavior of the derived estimator are compared with existing
estimators of the GMV portfolio. The resulting estimator shows significant
improvements and it turns out to be robust to the deviations from normality.
