Let $Y=X^\top W$ , with $X, W \in \mathbb{R}^{d \times d}$ are random matrices with standard normal entries. Let $\lambda_j$ be the $j^{th}$ singular value of $Y$. Is there a way to bound the tail probability of the $L_2$ norm of the top $k$ singular values of $Y$, i.e. $ P\Big(\sqrt{ \lambda_1^2(Y)+ \lambda_2^2(Y) + ... \lambda_k^2(Y)} > t \Big) \leq ? $ .

As a simple first try, I used $\sqrt{ \lambda_1^2(Y)+ \lambda_2^2(Y) + ... \lambda_K^2(Y)} \leq \sqrt{K}\lambda_1(Y)$ and used bounds for $\lambda_{max}$ of a matrix of the form of $Y$. But I need more tighter bounds and was wondering if there are other ways.

I would appreciate if you could let me know of any previous work done for this case that you might be aware of.