How to understand reproducibility in statistics? I'm having problems to find info about reproducibility in statistics. The books I have with an starting level in statistics doesn't even mention it.

At college, one of my previous professors explained it like something you just have to remember. She gave me a list of distributions with this property and the parameter that is involved with reproducibility. I don't like that because I have a really bad memory, in fact, I don't remember that list anymore. I think if I can understand properly why this happens and maybe reading a demonstration I will remember it much better and I will be able to use it faster.

I wonder if I'm using the wrong name for this property. I'm talking about:
$$
X_i\sim N(\mu,\sigma^2) \quad\forall i=1,\ldots,n \implies \sum X_i \sim N(n\mu,n\sigma^2)
$$
Some distributions only fulfill this property for one of its two parameters and some others doesn't fulfill it. I would like to know where is this property coming from.

Thanks
I haven't heard it called reproduceability, but the property you are referring to appears to be that of stable distributions:

independent sums of random variables from a stable distribution family will remain in that family, within a location-scale transformation.
I would call that "linearity" rather than "reproducibility".  A function f(x) of a single variable is linear if and only if f(ax+ by)= af(x)+ bf(y).