Suppose that zero-mean iid random matrices $A_1 ,A_2,\dotsc,A_n$ satisfy
$$\mathbb{P}\left(\left\|A_i \right\|\geq t\right)\leq \phi\left(t\right),\tag{*}$$
for $t>0$, where $\phi\left(t\right)$ is known (e.g., an appropriately decaying exponential or Gaussian function). Entries of each $A_i$ are not independent and they may be very correlated indeed.
I'd like to find a bound for $$\left\|\left[\array{A_1&A_2&\cdots&A_n}\right]\right\|$$ that holds with high probability.
Using (*) and the fact that $\left\|\left[\array{A_1&A_2&\cdots&A_n}\right]\right\|\leq \sqrt{\sum_{i=1}^n \left\|A_i\right\|^2}$ we can obtain such bound, but I wonder how sharp that bound would be. For instance, as an alternative we can bound the spectral norm of the matrix $\sum_{i=1}^nA_iA_i^\mathrm{T}$ using the Bernstein-type concentration inequalities for random matrices. This obviously yields a sharper bound, but is it possible to know that it would be substantially better than my naïve bound without going through the derivations?