Sum of truncated normal random variables

Question

It's known that the sum of two independent normal random variables is itself normal. Does this hold when dealing with the sum of two truncated normal random variables? I've seen this question, but it's not clear that the answer is exactly what I'm looking for here. Apologies if this has already been answered somewhere else.

Edit: I'm asking specifically about doubly truncated normal random variables with different bounds.

Answer 1

I'm aware this is not a solid answer, but it's too long for a comment. Even so, it sometimes helps to know whether to try verifying a counterexample or to try to constructing a proof. I'm pretty sure you should concentrate efforts on a counterexample. I do suspect that if both truncations are modest, the sum is "almost" a tuncated normal. (Maybe good enough for some practical purpose.)

1) Intuitive. Think about the probabilities in the tails of the sum. There will not be a clean cutoff as for a truncated normal, but a gentle fizzle.

2) Simulation. The R code below simulates a large sample of such sums and makes a histogram. It is an extreme case because the first of the standard normals is is very severely truncated to the interval (-.2, 2). It is impossible for me to imagine that the histogram of the sum could represent a truncated normal.

x = rnorm(10^6);  xt = x[abs(x)<.2];  y = rnorm(10^6);  yt = y[abs(y)<2]

xl = length(xt);  yl = length(yt)

# to make vectors of = length
ml = min(xl,yl) 

s = xt[0:ml] + yt[0:ml];  hist(s, breaks=20)