Let $A$ be a fixed $n$ by $n$ real symmetric positive definite matrix with eigenvalues $\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_n > 0$, and let $f(A):=\sum_{i=1}^n\log(\lambda_i)$, and let $X$ be a random $n$ by $k$ matrix with real iid copies distributed according to $N(0,\sigma^2/k)$.

The regime

• $n$ is fixed (in particular, $n \not \to \infty$).
• $k \to \infty$ (in particular, $k \gg n$).

Question

• How close is $f(A+XX^T)$ to $f(A)$ in espectation ?
• What is an upper-bound for $\mathbb P(|f(A+XX^T)-f(A)| > \epsilon)$?

Observations

I've observed that $f(A+XX^T)$ is approximately $\mathcal N(\mu,s^2)$, for some $\mu \in \mathbb R$, and $s > 0$.