= Cauchy sample mean has an unchanged distribution
{c}
{title2=$(X_1+\cdots+X_n)/n\sim\operatorname{Cauchy}(0,1)$}
= Cauchy sample mean stability
{c}
{synonym}
Independent standard <Cauchy distribution> variables sum to a Cauchy law of scale $n$, so their sample mean has its original distribution for every sample size. This follows by induction using <convolution of probability densities>. It does not contradict the usual <weak law of large numbers>, since the Cauchy law has no finite first absolute moment and its expectation is undefined. Symmetry does not turn a divergent expectation into zero.
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