Sample covariance matrices and their empirical spectral distributions

For positive integers $n$ and $p$, let $X\in {\mathbb{R}}^{n\times p}$ and define its (uncentered, unit-variance-normalized) sample covariance matrix by $S(X):=\frac{1}{n}{X}^{\mathsf{T}}X\in {\mathbb{R}}^{p\times p}\mathrm{.}$ It is real symmetric and non-negative. If its eigenvalues, counted with multiplicity, are ${\lambda }_1(S(X)),\dots ,{\lambda }_p(S(X))$, its empirical spectral distribution (ESD), normalized by the matrix dimension $p$, is ${\mu }_{S(X)}:=\frac{1}{p}{\sum }_{j=1}^p{\delta }_{{\lambda }_j(S(X))}\mathrm{.}$ The largest eigenvalue is ${\lambda }_{\max }(S(X)):={\max }_j{\lambda }_j(S(X))$. When $S(X)$ has at least one positive eigenvalue, its smallest positive eigenvalue is ${\lambda }_{\min }^{}(S(X)):=\min \{{\lambda }_j(S(X)):{\lambda }_j(S(X))>0\}$. For a sequence $(X_n{)}_{n\ge 1}$ with $X_n\in {\mathbb{R}}^{n\times p_n}$ and positive integer $p_n$ for every $n\ge 1$, write $S_n=S(X_n)$ and ${\mu }_n={\mu }_{S_n}$.

Bai and Silverstein, Chapter 3, §§3.1–3.3, printed pp. 39–58 (complete book PDF).

Junyi Liao, Chapter 2, §2.1, printed pp. 29–38 (complete lecture notes).