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Take $n\ll N$. Let $P$ be an $n\times N$ matrix of iid $\mathcal{N}(0,1)$ random variables, and let $D$ be an $N\times N$ diagonal matrix.

What can be said about the distribution of the largest singular value of $PD$?

When $n=1$, I get concentration inequalities from Lemma 1 of this paper. I would like a probabilistic upper bound for the general case, and I wouldn't be surprised if this is already available in the literature.

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I think the following answers your question (and more):

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