I'm trying to digest some litterature in time series analysis and don't quite understand what it means for a random variable to have finite second order moments. Tried to look it up in a probability theory textbook, however I don't quite get the intuition.
Thanks in advance!
The $k$-th moment of a random variable $X$ is $E(X^k)$. We say that this moment exists if $E(|X|^k)$ is a finite value (note the absolute value). If a r.v. has a finite $K$-th moment, the moments $k \le K$ are also finite. This property comes from the fact that the probability of the universe, the whole probability space, is $1$ and can be further justified by the comment by James Martin below this answer.
The variance is $Var(X) = E(X^2)-E(X)^2 = E(X^2)-E(X^1)^2 $. By the definition above and the property, if $X$ has a finite 2nd order moment, it has a variance (and conversely).