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

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.

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}
• Thanks for the reference! Although I believe there should be improvements based on the random nature of both $X$ and $Y$ Nov 18, 2022 at 23:08
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 $$Y^TY$$.