How does the spectrum of a product of $k$ random matrices behave around 0?
In particular, I'm wondering if the CDF of squared singular values behaves as $x^{\frac{1}{k+1}}$ around 0. The result for $k=1$ comes from Marchenko-Pastur.
Below is a visualization of empirical CDF of squared singular values of a matrix formed by multiplying together $k$ square $1000\times 1000$ matrices with standard Normal entries.
Unclear from simulation above is whether apparent "faster than power-law" decay is a finite sample effect or if it persists for $n\to\infty$. Any tips?
Motivation: a potential candidate for RMT model to match scaling laws observed in neural network training (background). Shape of loss curve over time for $t\to \infty$ comes from shape of spectrum for $x\to 0$