I read authors use the notion of "moment condition" and equivalently stating it is $E|X|<\infty$. Some have used the notion of "regularity condition" and try to explain why it is reasonable to assume or impose moment conditions in large sample theory.
I have the following questions:
(0) What is the difference between $E|X|<\infty$ and $E(X)<\infty$?
(1) What does this moment condition mean?
(2) Why is it important to have this so-called "regularity" condition in asymptotic analysis? I understand to the extent that it is required in both the Strong and Weak Law of Large Numbers theorems.
(3) Why is it sometimes equivocated to be "regularity" condition, and does this relate to the regularity terms we hear from undergrad Calculus class?
(4) Where else is this used in statistical inference or other than asymptotic analysis? Perhaps the discussion of similar moment conditions in the generalized method of moments or maximum likelihood method?
Thanks!