@IosifPinelis Agreed. In fact with thick tails (unbounded variance) I would surprised if one would get even a $1/\sqrt n$ convergence.

I wonder if one can use Markov inequality to demonstrate that no matter where you 'place' the mean there is much more mass in the tail than Markov inequality would indicate. These infinitely many hypohesis needs to be accounted for, perhaps with VC dimension arguments.

@IosifPinelis Thanks a lot for adding the citations. I have been reading up on Hill estimators and their likes the past few months. It seems the question of how best to estimate where the tail starts (needed in the computation of the estimators) hasn't been satisfactory settled yet.

I understood this intuitively at the time of posing the question, so I appreciate your time to make it rigorous. I should have framed it better. What I had in mind was -- given that I need to distinguish between the bounded and unbounded expectations how much can one weaken the setting from the easy case of a distinguishing between two parametric distributions of finite and infinite expectations. For example growth rate on $E[X^\alpha]$ for $\alpha \uparrow1$, or unimodality, or monotonicity or smoothness etc. There must be literature on such criteria, would appreciate pointers

Isnt it reasonable to keep the distribution independent of the number of samples. But of course one should be able to construct cases even with that restriction.

@MattF. I see, so with that method I need to be extremely lucky for it to work, way too many parametric families and that still will not cover all distributions. Any family that forms a net for distributions with infinite expectation ?

@MattF. But isn't Cauchy one of the infinitely many ways expectation can be unbounded, would testing for Cauchy test still be appropriate if the true distribution is different ?