Note: A previous question asking about the finiteness of second moments was asked in 2010 -- this question is not a duplicate of that one because (1) this question is about tests for the finiteness of arbitrary moments, including the first moment (2) all of the papers I have found regarding this question were published after 2010, again, when the other question was asked.
Question: Does anyone know what the state of current literature is for hypothesis tests regarding the finiteness of moments of empirical distributions?
I have found at least three papers with Google, (1)(2)(3), all published after 2010, which claim to describe tests to address this question. However, the accepted answer to the previous question claims that any such test is impossible. So are all three papers wrong?
(Considering how often incorrect papers are published, I don't consider that a remote possibility.)
Motivation/Background: The Strong Law of Large Numbers requires finite first moments, and the Central Limit Theorem requires finite second moments. These results are crucial to the validity of most statistical tests used in practice, as far as I am aware. However, I do not know of any widely used methods to test the validity of these assumptions in empirical datasets. This is especially concerning to me since the typical real-valued continuous probability distribution does not have finite first moments, and of the (atypical) real-valued continuous probability distributions which do have finite first moments, most in turn do not have finite second moments.
Thus the state of the art of such hypothesis-testing procedures seems to be absolutely essential to the correct practice of statistics, unless for example a concerted effort is made to change all statistical tests used in practice to ones with the default assumption that no moments are finite.