Linear combination of two random non-normals that is still a member of same family It is well-known that a linear combination of 2 random normal variables is also a random normal variable.  Are there any common non-normal distribution families (e.g., Weibull) that also share this property? There seem to be many counterexamples.  For instance, a linear combination of uniforms is not typically uniform.  In particular, are there any non-normal distribution families where both of the following are true:


*

*A linear combination of two random variables from that family is equivalent to some distribution in that family.

*The resulting parameter(s) can be identified as a function of the original parameters and the constants in the linear combination.  


I'm especially interested in this linear combination:
$Y = X_1 \cdot w + X_2 \cdot \sqrt{(1-w^2)}$
where $X_1$ and $X_2$ are sampled from some non-normal family, with parameters $\theta_1$ and $\theta_2$, and $Y$ comes from the same non-normal family with parameter $\theta_Y = f(\theta_1, \theta_2, w)$. 
I'm describing a distribution family with 1 parameter for simplicity, but I'm open to distribution families with multiple parameters.
Also, I'm looking for example(s) where there is plenty of parameter space on $\theta_1$ and $\theta_2$ to work with for simulation purposes.  If you can only find an example that works for some very specific $\theta_1$ and $\theta_2$, that would be less helpful.
 A: It is well-known that a linear combination of 2 random normal variables is also a random normal variable. Are there any common non-normal distribution families (e.g., Weibull) that also share this property? 
The normal distribution satisfies a nice convolution identity: $X_1\sim N\left[\mu _1,\sigma _1^2\right],X_2\sim N\left[\mu _2,\sigma _2^2\right]\Longrightarrow X_1+X_2\sim N\left[\mu _1+\mu _2,\sigma _1^2+\sigma _2^2\right]$. If you are referring to the central limit theorem, then for example, those gamma distributions with the same shape coefficient would share that property and convolve to be gamma distributions. Please see A cautionary note regarding invocation of the central limit theorem. In general, however, with unequal shape coefficients, gamma distributions would "add" by a convolution that would not be a gamma distribution but rather a gamma function multiplying a hypergeometric function of the first kind as found in Eq. (2) of convolution of two gamma distributions. The other definition of adding, that is forming a mixture distribution of unrelated processes would not necessarily exhibit any central limit, for example, if the means are different.
There are probably other examples, I haven't done an exhaustive search. Closure for convolution does not seem to be far fetched. For linear combination, the product of Pearson VII with a Pearson VII is another Pearson VII.
A: 
It is well-known that a linear combination of 2 random normal variables is also a random normal variable. Are there any common non-normal distribution families (e.g., Weibull) that also share this property?

I sounds like you are looking for the class of Levy-stable distributions.  This is the class $\mathscr{P}$ of all distributions $\mathcal{P} \in \mathscr{P}$ that satisfy the stability property:
$$X_1, X_2, X_3 \sim \text{IID } \mathcal{P} \quad \quad \implies \quad \quad (\forall a) (\forall b) (\exists c>0) (\exists d): \ aX_1 + bX_2 \overset{\text{Dist}}{\sim} c X_3 + d.$$
In other words, for every distribution in this class, if you take a linear function of two independent random variables with that distribution, then this has the same distribution as an affine function of a single random variable with that distribution.  (Note that this stability requirement can be tightened by setting $d=0$, which gives the subclass of strictly stable distributions.)
The Levy-stable distributions can be considered to be a family of distributions in its own right, and in this sense it is the only family of distributions with this stability property, since (by definition) it encompasses all distributions with this property.  The normal distribution falls within the class of Levy-stable distributions, as does the Cauchy distribution, the Landau distribution, and the Holtsmark distribution.
