# What are the parameters of the sub-exponential distribution that is sub-Gaussian squared?

I understand that the square of a sub-Gaussian is sub-exponential (https://arxiv.org/pdf/1011.3027.pdf, Lemma 5.14) but it's not clear to me as to what the parameters of the sub-exponential are, given that the sub-Gaussian has parameter $$\sigma$$.

I know that a Gaussian with parameter $$\sigma$$ when squared yields a sub-exponential with parameters $$(2\sigma^2,4\sigma^2)$$, would the same apply to a sub-Gaussian too?

Answering my own question, this paper shows that the parameters of the sub-exponential are $$(4\sqrt{2} \sigma^2, 4 \sigma^2)$$ http://proceedings.mlr.press/v33/honorio14-supp.pdf (although I do not have a tighter characterization of the parameters).