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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?

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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).

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