My question concerns the proof of Lemma 1 in this blog post of Terence Tao.
In the first paragraph, he says:
Applying standard concentration of measure results (e.g. Exercise 4, Exercise 5, or Theorem 9 from Notes 1), we see that each ${X_i \cdot x}$ is uniformly subgaussian.
Here, the matrix entries are assumed to be bounded, centered random variables. Could we proceed as follows, not using concentration of measure and instead working more generally in the case where the entries are merely sub-Gaussian? (Note that a bounded, centered random variable is sub-Gaussian.)
Each entry is centered and sub-Gaussian, so any linear combination of them will also be sub-Gaussian. In particular, since $x$ is fixed, $x\cdot X_i$ is sub-Gaussian. Then we recover the tail estimate Tao gives, and the rest of the proof goes through as before.
If this works, it would solve Exercise 3 by generalizing Corollary 4 to entries with sub-Gaussian tails.
I would also be interested in any other solutions to Exercise 3 that might exist. In particular, I suspect some sort of truncation argument will work, but I can't find it.