Let $X \sim \mathcal{N}(0, \sigma^2)$ be a Gaussian random variable. Is there a name or any properties of note for the random variable $Y = I_0(X)$, where $I_0$ is the Bessel function of order $0$ evaluated at $X$?
$I_0$ occurs as a component of the density function for several probability distributions. I am trying to understand how the Gaussian RV is transformed under the action of the Bessel function.