  This text is devoted to the asymptotic study of some spectral properties of
the Gram matrix $W^{\sf T} W$ built upon a collection $w_1, \ldots, w_n\in
\mathbb{R}^p$ of random vectors (the columns of $W$), as both the number $n$ of
observations and the dimension $p$ of the observations tend to infinity and are
of similar order of magnitude. The random vectors $w_1, \ldots, w_n$ are
independent observations, each of them belonging to one of $k$ classes
$\mathcal{C}_1,\ldots, \mathcal{C}_k$. The observations of each class
$\mathcal{C}_a$ ($1\le a\le k$) are characterized by their distribution
$\mathcal{N}(0, p^{-1}C_a)$, where $C_1, \ldots, C_k$ are some non negative
definite $p\times p$ matrices. The cardinality $n_a$ of class $\mathcal{C}_a$
and the dimension $p$ of the observations are such that $\frac{n_a}{n}$ ($1\le
a\le k$) and $\frac{p}{n}$ stay bounded away from $0$ and $+\infty$. We provide
deterministic equivalents to the empirical spectral distribution of $W^{\sf
T}W$ and to the matrix entries of its resolvent (as well as of the resolvent of
$WW^{\sf T}$). These deterministic equivalents are defined thanks to the
solutions of a fixed-point system. Besides, we prove that $W^{\sf T} W$ has
asymptotically no eigenvalues outside the bulk of its spectrum, defined thanks
to these deterministic equivalents. These results are directly used in our
companion paper "Kernel spectral clustering of large dimensional data", which
is devoted to the analysis of the spectral clustering algorithm in large
dimensions. They also find applications in various other fields such as
wireless communications where functionals of the aforementioned resolvents
allow one to assess the communication performance across multi-user
multi-antenna channels.
