Sum of normal truncated random variables Suppose I have $n$ independent normal random variables
$$X_1 \sim \mathrm{N}(\mu_1, \sigma_1^2)\\X_2 \sim \mathrm{N}(\mu_2, \sigma_2^2)\\\vdots\\X_n \sim \mathrm{N}(\mu_n, \sigma_n^2)$$
and $Y=X_1+X_2+\dotsm+X_n$. How would I characterize the density of $Y$ if the distribution of each $X_i$ is each truncated to within $(\mu_i - 2\sigma_i, \mu_i + 2\sigma_i)$? In other words, I'm sampling from $n$ independent normal distributions, discarding samples not within $2\sigma_i$ of each mean, and summing them.
Right now, I'm doing this with the R code below:
x_mu <- c(12, 18, 7)
x_sd <- c(1.5, 2, 0.8)
a <- x_mu - 2 * x_sd
b <- x_mu + 2 * x_sd

samples <- sapply(1:3, function(i) {
  return(rtruncnorm(100000, a[i], b[i], x_mu[i], x_sd[i]))
})

y <- rowSums(samples)

Is there any method for generating the density of $Y$ directly?
 A: You could use approximation by saddlepoint methods, for the sum of truncated normals. I will not give the details now, you can look at my answer to  General sum of Gamma distributions   for hints. What we need is to find the moment-generating function for a truncated normal, which is easy. I will do it here for a standard normal truncated at $\pm 2$, which has density
$$
   f(x) =\begin{cases}  \frac1{C} \phi(x), & |x| \le 2 \\
                                    0,     & |x| > 2 \end{cases}
$$
where $C=\Phi(2) - \Phi(-2)$ here $\phi(x), \Phi(x)$ are density and cdf for a standard normal, respectively. 
The moment generating function can be calculated as
$$ \DeclareMathOperator{\E}{\mathbb{E}}
M(t) = \E e^{tX}=\frac1{C}\int_{-2}^2 e^{tx} \phi(x)\; dx=\frac1{C}e^{\frac12 t^2} [\Phi(2-t)-\Phi(-2-t) ]
$$
and then we can use saddlepoint approximations.
A: I'm curious why, but yes, there is a simple way to generate the pdf of this sum of distributions:
## install.packages("truncnorm")
## install.packages("caTools")
library(truncnorm)

x.mu <- c(12, 18, 7)
x.sd <- c(1.5, 2, 0.8)
x.a <- x.mu - 2*x.sd
x.b <- x.mu + 2*x.sd

dmulti <- function(x, a, b, mu, sd)
  rowSums(
    sapply(1:length(mu),
           function(idx)
             dtruncnorm(x, a=a[idx], b=b[idx], mean=mu[idx], sd=sd[idx])))/length(mu)
pmulti <- function(q, a, b, mu, sd)
  rowSums(
    sapply(1:length(mu),
           function(idx)
             ptruncnorm(q, a=a[idx], b=b[idx], mean=mu[idx], sd=sd[idx])))/length(mu)

pointrange <- range(c(x.a, x.b))
pointseq <- seq(pointrange[1], pointrange[2], length.out=100)
## Plot the probability density function
plot(pointseq, dmulti(pointseq, x.a, x.b, x.mu, x.sd),
     type="l")

## Plot the cumulative distribution function
plot(pointseq, pmulti(pointseq, x.a, x.b, x.mu, x.sd),
     type="l")

