I'd like to ask for additional info regarding a previous post on the subject:
but I can't comment directly on that.
Assume $X \sim N(\mu_{1}, \sigma_1^2)$ is doubly truncated (below and above) at ($a,b$)
and
$Y \sim N(\mu_{2}, \sigma_2^2)$ is doubly truncated (below and above) at ($a,b$).
$X$ and $Y$ have the same bounds $(a, b)$ and are independent.
Can the density of $Z=X+Y$ be reasonably approximated by a truncated Gaussian with $\mu_{3} = \mu_{1} + \mu_{2}$, $\sigma_3^2=\sigma_1^2+\sigma_2^2$, and bounds given by $(2a,2b)$?