How to obtain tail bounds for a square of sub-Gaussian random variable?

Solution 1:

If $Z$ is subgaussian, then $Z^2$ is known (somewhat confusingly) as a subexponential random variable. (I say "somewhat confusingly" since in other subfields of probability theory that term means something else entirely).

Have you looked through Roman Vershynin's notes on nonasymptotic random matrix theory? Here's the paper:

R. Vershynin, Introduction to the non-asymptotic analysis of random matrices, Aug. 2010.

There are also some lecture notes that are older on which I believe this paper was based. One version of them can be found here.

This may not contain exactly what you're looking for, but I suspect they'll be helpful.

You might also look through the publications of Mark Rudelson, who also works on topics related to the types of concentration inequalities you are interested in.