A zero mean sub-Gaussian random variable $Z$ satisfies ${\mathbb E} \exp(tZ) \leq \exp(t^2\sigma^2/2)$ for some constant $\sigma > 0$. This bound can be used together with the Chernoff bound to obtain a two sided tail bound.
I am interested in obtaining exponential tail bounds on $\mathbb{P}[Z^2 - \mathbb{E}Z^2 > z] $ and $\mathbb{P}[\mathbb{E}Z^2 - Z^2 > z]$. One difficulty is in obtaining an upper bound on $\mathbb{E}\exp[t(Z^2 - \mathbb{E}Z^2)]$. Any pointers to relevant literature? This seems like a result that should be known.
If $Z$ is a standard Gaussian random variable, then $Z^2$ is distributed according to the central chi-squared distribution and the above probabilities can be bounded as in Laurent and Massart (2000) -- Adaptive estimation of a quadratic functional by model selection.