Stability of Gaussian random variables

$\mathcal{N}(\mu_1,\sigma_1^2)+\mathcal{N}(\mu_2,\sigma_2^2)=\mathcal{N}(\mu_1+\mu_2,\sigma_1^2+\sigma_2^2)$

Refer to Gaussian concentration, Gaussian tail bound.

See: stable distribution, i.e. Lévy alpha-stable distribution

References:

1. https://en.wikipedia.org/wiki/Stable_distribution