Typical (or common) approaches to prove an estimator is consistent require finite mean and variance. The proofs usually follow from concentration bounds, e.g. Markov, Chebyshev, etc.
I'm wondering how one shows consistency of an estimator if the expectation is infinite?
Or alternately-and leading to a similar conclusion but perhaps more general-are there concentration of measure bounds when the mean is infinite? (please provide example, proof and/or links)