I'm trying to get my head around the concept of "stable distribution" but there is a concept which is not clear to me. The thing that I would like to understand is whether
"Undefined variance" is equivalent to "infinite variance"
At the beginning, I thought that the two concepts where equivalent statements but then I looked on wikipedia the page about levy distribution (https://en.wikipedia.org/wiki/Lévy_distribution) and I saw that
For the Lèvy distribution:
Variance = $\infty \qquad$ Kurtosis=Undefined
Hence, I would like to understand the difference between an undefined moment and an infinite one
Wikipedia say the Cauchy distribution has an undefined variance. This makes sense in the sense that it also has an undefined mean, and so considering the expected square of the deviation from the mean is not meaningful.
Personally I would probably say the same about the Lévy distribution, which has an infinite mean. But there is an argument that since the Lévy distribution is almost surely finite, the expected square of the deviation from this infinite mean would then also be infinite; something similar could be said about the absolute value of a Cauchy random variable or a log-Cauchy random variable, which both have support on the positive real line and infinite means
A simpler case of infinite variance being a meaningful statement would be a Student's t-distribution with $2$ degrees of freedom. The mean is $0$ and so it is sensible to talk about the expected square of the deviation from the mean, and to find that it is infinite
In all three of those examples, the second moment about zero is infinite, as is the second moment about any finite value. The distinction between undefined variance and infinite variance is whether the deviation from the mean is a meaningful concept