When I tried to get the convergence rate of $\|\hat{\Omega}\hat{\Sigma}-\Omega\Sigma\|_2$, I found that I have to face the term $\|\Omega-\hat{\Omega}\|_2\|\Sigma-\hat{\Sigma}\|_2$ which is the product of the concentration inequality. that's because $$ \|\hat{\Omega}\hat{\Sigma}-\Omega\Sigma\|_2 \le \|\Omega-\hat{\Omega}\|_2\|\Sigma-\hat{\Sigma}\|_2+\|\Omega-\hat{\Omega}\|_2\|\Sigma\|_2+\|\Sigma-\hat{\Sigma}\|_2\|\Omega\|_2. $$
I could get the convergence rate of $\|\Omega-\hat{\Omega}\|_2 \le C_1\sqrt{\frac{p}{n}}$ and $\|\Sigma-\hat{\Sigma}\|_2\le C_2\sqrt{\frac{p}{n}}$ separately, but the product of two estimators makes me confused. Both $\sqrt{\frac{p}{n}}$ and $\frac{p}{n}$ will appear in the final result at the same time. But this seems to be rare in papers. Is this correct? I guess only $\sqrt{\frac{p}{n}}$ can appear in the final convergence rate because $\frac{p}{n}$ could be bounded by $\sqrt{\frac{p}{n}}$。
