I am trying to bound the following expression $$||\frac{1}{N} W^TQW - \lambda \mathbf{1}_N||_F~,$$ where $W\in\mathbb{R}^{N\times m }$ with $||W||_{\text{op}} \leq \sqrt{N} C'$, $Q\in\mathbb{R}^{N\times N }$ a diagonal matrix with $||Q||_{\text{op}} \leq \sqrt{N} C$, $\lambda = \frac{1}{N} \text{tr} Q$ and $m/N\to 0$ for $N\to\infty$. I know that $\tfrac{1}{\sqrt{N}}||W - W_0||_F\leq cN^{-1/2}$, where the entries $(W_0)_{ij}$ of the reference matrix $W_0$ are iid random variables drawn from a normal distribution $\mathcal{N}(0,1)$.
My unsuccessful approach was $$||\frac{1}{N} W^TQW - \lambda \mathbf{1}_N||_F\leq ||\frac{1}{N} W_0^TQW_0 - \lambda \mathbf{1}_N||_F + ||\frac{1}{N} W^TQW - \frac{1}{N} W_0^TQW_0||_F~,$$
The first part can be bounded with high probability using Markov inequality. I tried to bound the second term as follows: $$ ||\frac{1}{N} W^TQW - \frac{1}{N} W_0^TQW_0||_F \leq 2 ||Q||_{op} \tfrac{1}{\sqrt{N}}||W||_{op} \tfrac{1}{\sqrt{N}}||W - W_0||_F~.$$ However, the last expression on the right hand side above does not go to zero because $||Q||_{op}\tfrac{1}{\sqrt{N}}||W - W_0||_F$ can be bounded to be less or equal to a constant $cC$.
I am puzzled, because in the expression to be bound the dimension $m$ is fixed and does not go to $\infty$, so my assumption is that it should go to zero with high probability as well.
Does somebody have an idea what property to analyze or how to proceed?
Thanks in advanced!
Update I the meantime I played with the expressions above. I still believe the bound holds, however, the presented bound and estimates may be not tight enough to proof the desired result. In case that I will find a solution to the problem in future I let you know.