Suppose that $X \sim N(\mu, \sigma^2)$ is Gaussian. Then $Y := |X|$ is a Folded normal random variable.
$X$ is clearly Subgaussian i.e. $M_{X}(\lambda) := E\left(e^{\lambda(X - \mu)}\right) \leq e^{\sigma^{2} \lambda^{2} / 2}, \; \forall \lambda \in \mathbb{R}$. Where $M_{X}(\lambda)$ is the moment generating function of $X$.
Is $Y$ also Subgaussian? And if so with what parameters?
Based on the linked page for the Folded normal, it appears that $M_{Y}(t) = e^{\frac{\sigma^{2} t^{2}}{2}+\mu t} \Phi\left(\frac{\mu}{\sigma}+\sigma t\right)+e^{\frac{\sigma^{2} t^{2}}{2}-\mu t} \Phi\left(-\frac{\mu}{\sigma}+\sigma t\right)$. I'm not sure how to use this, since the Folded normal has a much more complicated mean.
Could anyone please provide guidance on how to verify this property?