# Non-asymptotic ratio of singular values for random matrices

Suppose we have two independent Gaussian random vectors $\mathbf{X}\in \mathbb{R}^{p\times 1}, \mathbf{X}\sim \mathcal{N}(\mathbf{0}, \Sigma_x)$ and $\mathbf{Y}\in \mathbb{R}^{q\times 1}, \mathbf{Y}\sim \mathcal{N}(\mathbf{0}, \Sigma_y)$. We draw a sample of size $n$ from both random vectors, say $(\mathbf{x}_i, \mathbf{y}_i)$ for $i = 1, \ldots, n$, and compute the following matrix:

$$S_n = \frac{1}{n}\sum_{i=1}^{n} \mathbf{x}_i\mathbf{y}_i^{\top}.$$

The quantity that we are interested is the fraction of variance stored in the first singular value of $S_n$; i.e. we want to estimate the following ratio:

$$r_n := \frac{\sigma_1}{\sum_{i=1}^{\min(p, q)} \sigma_i},$$ where $\sigma_1 \geq \sigma_2\geq \ldots \geq\sigma_{\min(p, q)}\geq 0$ are the singular values of the matrix $S_n$.

How can we estimate $r_n$ in non-asymptotic regime ($n < \infty$) or at least find bounds for it? The matrix dimensions $p$ and $q$ are allowed to go to infinity.

-