Concentration inequality for the spectral norm of the product of normalized Gaussian (and subgaussian) matrices in high dimensions

Question

Let $X,Y$ be two $n\times n$ i.i.d. Gaussian matrices (entries are i.i.d N(0,1) and $X$ and $Y$ are independent).

Consider their product normalized by the standard variance of entries $\frac{XY}{\sqrt n}$. I wonder if there are any results giving estimates for the spectral norm of the matrix $\frac{XY}{\sqrt n}$ around its mean with high probability as $n \to \infty$.

As a side question, I also wonder if we could have estimates for the analogous problem for sub-Gaussian matrices (entries are i.i.d sub-Gaussian). Again we require the product $XY$ be normalized by the standard variance of entries.

The main difficulty for this problem is that the entries in the matrix $XY$ are no longer i.i.d.

Answer 1

Vershynin, R. Spectral norm of products of random and deterministic matrices. Probab. Theory Relat. Fields 150, 471–509 (2011).

This result is a sharp bound on the spectral norm of $W=BA$, where $A$ is random with independent and centered entries and $B$ is deterministic. It is assumed has the random matrix has finite $(4 + \epsilon)$ moment, which is implied by your sub-Gaussian assumption.

While the result of the paper is for deterministic $B$, this actually applies to your case because we can bound $$ P \left (n^{-1/2} \left \vert \left \Vert XY \right \Vert - \mathbb{E}\left \Vert XY \right \Vert]\right \vert \vert > t \right \vert X) \leq f(n), $$ so by smoothing, law of total expectation, tower law (whatever you want to call it), \begin{align} P \left (n^{-1/2} \left \vert \left \Vert XY \right \Vert - \mathbb{E}\left \Vert XY \right \Vert]\right \vert \vert > t \right) & = \mathbb{E} \left [P \left (n^{-1/2} \left \vert \left \Vert XY \right \Vert - \mathbb{E}\left \Vert XY \right \Vert]\right \vert \vert > t \right \vert X) \right ] \\ & \leq \mathbb{E} f(n) \\ & = f(n). \end{align}

Answer 2

The Marcenko-Pastur resut (see, e.g., https://www.sciencedirect.com/science/article/pii/S0047259X85710512) gives you the Stieljes equation of the limiting spectral distributions of matrices of the form $X^T T X$ (properly normalized) where $X$ has iid entries and $T$ is diagonal and has a limiting spectral distribution. Then the Stieljes equation of the limit depends on the spectral distribution of $T$. In your case, by rotational invariance of $X$ and independence you can set $T$ to be diagonal with elements the eigenvalues of $Y^TY$.