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:


*

*what does independent copies of a random variable mean?

*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?
 A: The central limit theorem and infinite divisibility are properties associated with the normal distribution.  The stable distributions extend family of distributions to other distributions that have the inifinite divisibility property and and a limit theroem called the stable law.  Convergence of avergaes to the normal distribution requires certain properties of the random variables.  We shall just consider the simple case where the random variables are independent and identically distributed. in that case the central limit theorem holds under mild conditions. (1) the variance must exist and (2) any moment slightly higher than the 2nd moment must exist (one form of the CLT).  Now distributions that do not have a finite variance and even some like the Cauchy that don't have a finite mean will still have their independent sums properly normalized converge to one of the stable distributions.  The stable distributions are indexed by a parameter alpha that can be >0 but <=2.  The index 2 corresponds to the normal distribution. A more thorough account of the properties of stable distributions can be found in Wikipedia at this link: 
http://en.wikipedia.org/wiki/Stable_distribution 
The other questions were answered correctly by VitalStatistix in his comments.
