What is the difference between infinitely divisible and stable law? I read somewhere that stable law is the special case of infinitely divisible. In other word, stable distribution is a special case of infinitely divisible distribution. 
But I am not quite sure what differentiate both of them apart.  
 A: Every stable law is infinitely divisible, but some infinitely divisible laws are not stable.  An example of the latter is the Poisson distribution.  For each $\lambda > 0$ and each $n\in\{1,2,3,\ldots\}$, there exist independent random variables $X_1,\ldots,X_n$ each distributed as $\mathrm{Poisson}(\lambda/n)$, so that $X_1+\cdots+X_n\sim\mathrm{Poisson}(\lambda)$.  Thus each Poisson distribution is infinitely divisible.  But the Poisson distribution is not stable, since, if you add two independent identically distributed random variables $X$ and $Y$, each with a Poisson distribution, what you get does not have the same distribution as $\alpha X+\beta$ for some $\alpha,\beta\in\mathbb R$.
A: If a random variable $X$ is infinitely divisible, then for each $n$ we can write $$X = Y_1 + \cdots + Y_n$$
for some i.i.d. random variables $Y_i$.  (Infinitely is a bit of a misnomer here -- all we really mean is that $n$ can be arbitrarily large.)
If $X$ is stable, then we can moreover do this in such a way that the $Y_i$ have the same distribution as $X$, up to scaling.
