Let $a \in \mathbb R$ be a determinstic scalar and let $X$ be and $n \times d$ such that the $n \times n$ psd random matrix $S=XX^T$ has limiting eigenvalue distribution $\mu$, when $n,d \to \infty$ with $n/d = \gamma \in (0,\infty)$. Consider the psd random matrix $T=YY^\top$, where $Y$ is the $n \times d$ random matrix with entries $Y_{ij} = X_{ij} + a$.

Question. What is the limiting eigenvalue distribution $\lambda$ of $T$ ?

Note. My guess is that the Stieltjes transform of $\lambda$ would be some how given implicitly via the Stieltjes transform of $\mu$.

The example I have in mind is when $X$ has iid rows drawn from an isotropic log-concave distribution in $\mathbb R^d$. For gaussian of uniform on unit-sphere in $\mathbb R^d$.

Let $a \in \mathbb R$ be a determinstic scalar and let $X$ be and $n \times d$ such that the $n \times n$ psd random matrix $S=XX^T$ has limiting eigenvalue distribution $\mu$, when $n,d \to \infty$ with $n/d = \gamma \in (0,\infty)$. Consider the psd random matrix $T=YY^\top$, where $Y$ is the $n \times d$ random matrix with entries $Y_{ij} = X_{ij} + a$.

Question. What is the limiting eigenvalue distribution $\lambda$ of $T$ ?

The example I have in mind is when $X$ has iid rows drawn from an isotropic log-concave distribution in $\mathbb R^d$. For gaussian of uniform on unit-sphere in $\mathbb R^d$.

Notes

• My guess is that the Stieltjes transform of $\lambda$ would be some how given implicitly via the Stieltjes transform of $\mu$.
• I'm only interesting in being able to integrate w.r.t $\lambda$. For example, I'm interested computing sums like $\sum_i \dfrac{\ell_i}{(\ell_i + \lambda)^2}$, where $\ell_1 \ge \ell_2 \ge \ldots $ are the eigenvalues of $T$.