= IID does not imply finite moments
{c}
The <IID sample> property places no integrability condition on its common <probability distribution>. An <IID sample> with the standard <Cauchy distribution> has no finite <expected value> or <variance>. The <Cauchy sample mean stability> result shows that its averages keep that same <Cauchy distribution> for every sample size. Thus increasing $n$ need not concentrate an <IID sample> around a population <mean> that does not exist. The finite-variance bound below applies only when its hypotheses hold.
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