  We analyze the complexity of sampling from a class of heavy-tailed
distributions by discretizing a natural class of It\^o diffusions associated
with weighted Poincar\'e inequalities. Based on a mean-square analysis, we
establish the iteration complexity for obtaining a sample whose distribution is
$\epsilon$ close to the target distribution in the Wasserstein-2 metric. In
this paper, our results take the mean-square analysis to its limits, i.e., we
invariably only require that the target density has finite variance, the
minimal requirement for a mean-square analysis. To obtain explicit estimates,
we compute upper bounds on certain moments associated with heavy-tailed targets
under various assumptions. We also provide similar iteration complexity results
for the case where only function evaluations of the unnormalized target density
are available by estimating the gradients using a Gaussian smoothing technique.
We provide illustrative examples based on the multivariate $t$-distribution.
