# Interpreting definition of stable distributions

I am trying to interpret the following definition:

A non-degenerate distribution is a stable distribution if it satisfies the

following property:
Let X1 and X2 be independent copies of a random variable X. Then X is said to

be stable if for any constants a>0 and b>0 the random variable aX1 + bX2 has

the same distribution as cX + d for some constants c>0 and d. The distribution

is said to be strictly stable if this holds with d = 0 (Nolan 2009).


I have searched the internet for more intuitive explanations but couldn't find anything. Can someone help?

I couple of questions:

1. what does independent copies of a random variable mean?

2. is the random variable X already a distribution?

So does this say basically that if we add for example two normal distributions then it will also be normal?

• Any random variable, by definition, has a distribution, which might be discrete or continuous based on the set of values it admits and independent copies mean independent draws from the distribution of $X$, i.e. $X_1,X_2 \sim F_X$. Stable distributions are called so because a sum such as the one you described, leaves the shape of the distribution unchanged. Also, the example with Normal is correct. i.e. $X \sim N(\mu,\sigma^2) \Rightarrow aX_1+bX_2 \sim N((a+b)\mu, (a^2+b^2)\sigma^2)$ and $cX+d \sim N(c\mu+d, c^2)$, so, the definition holds for $c^2 = a^2+b^2$ and $d = (a+b-c)\mu$. Commented Jul 17, 2012 at 22:08