TITLE: PDF of summation of independent random variables with different mean and variances
QUESTION [5 upvotes]: What can we say about the probability distribution function of $n$ independent random variables with different means and variances but the same PDF?
For example, lets say $X_1,X_2,\dots,X_n$ are independent random variables with the following PDFs:
$$
\begin{align}
X_1 & \sim N(μ_1,σ_1) \\
X_2 & \sim N(μ_2,σ_2) \\
& {}\,\vdots \\
X_n & \sim N(μ_n,σ_n)
\end{align}
$$
So they are all Normal random variables with different means and variances and their summation would also have Normal distribution. Now my question is what about other distribution functions?

REPLY [2 votes]: It seems the question is to find families $F$ of distributions, other than the gaussian family, such that the distribution of every linear combination of independent random variables with distributions in $F$ is in $F$. The problem is not really well defined hence let us specify things somewhat:

Let $F_h$ denote the family of distributions of random variables $X$ such that $E[\mathrm e^{\mathrm itX}]=\mathrm e^{\mathrm it\mu-h(\sigma t)}$ for some $(\mu,\sigma)$ with $\sigma\geqslant0$. Does it satisfy the property above?

Note that the gaussian case is when $h(t)=\frac12t^2$. The family $F_h$ satisfies the property above if and only if, for every $\sigma_1$ and $\sigma_2$, there exists some $\mu$ and $\sigma$ such that
$$
h(\sigma_1 t)+h(\sigma_2 t)=h(\sigma t)+\mu t.
$$
In particular, stable distributions (for some fixed exponent $\alpha$) spring to mind.