CLT and stable distributions I have a few questions about generalizations of the CLT and stable distributions. I'm trying to correct my understanding and make it precise. Please forgive my naivete, I am not a professional statistician :-)
If I take the sum of a large enough sequence of independent R.V.'s, do they always converge to a stable distribution? (I've heard about generalizations of the CLT, but I'm looking for more precision).
When working with real data, what would be a hint that I need to model with a stable distribution? Is it possible to perform max likelihood with stable distributions?
 A: No there are distributions that do not satisfy the conditions to be in the domain of attraction of a stable law.
Theorem 2(a) Feller "An Introduction to Probability Theory and Its Applications Volume II page 577: In order that a distribution F belongs to some domain of attraction it is necessary that the truncated moment function μ(x) varies regularly with an exponent $2-α$, $(0<α<=2)$.
$μ(x) =∫y^2 dF(y)$ where the limits of integration are from $-x$ to $x$. 
$μ(x)$ varies regularly means $μ(x) ~ x^{2-α} L(x)$ where $L(x)$ is a slowly varying function and slowly varying means $L(tx)/L(t) → 1$ as $t→∞$.
A: 
If I take the sum of a sequence of independent R.V.'s, do they always converge to a stable distribution? (I've heard about generalizations of the CLT, but I'm looking for more precision).

I think we need more restrictions on this statement to say anything useful. You could have a sequence of independent random uniform RVs on the interval [0, i]. The convolution of any two RVs from this sequence certainly does not follow the same distribution and the asymptotic distribution of the sample mean converges to a non-stable distribution.
A: As far as max likelihood, start with this: the Gaussian distribution is stable, so certainly it is possible in some circumstances.
