= Solution
Let $(Y_n)$ be <independent and identically distributed random variables> with the standard <Cauchy distribution>, whose <probability density function> is
$$
f(y)=\frac{1}{\pi(1+y^2)}.
$$
Its first absolute moment diverges since
$$
\mathbb E|Y_1|=\frac{2}{\pi}\int_0^\infty\frac{y}{1+y^2}\,dy=\infty.
$$
On the other hand, its <characteristic function> is $\varphi(t)=e^{-|t|}$. The <characteristic function of a sum of independent variables> therefore gives
$$
\mathbb E\exp\left(it\frac{S_n}{n}\right)
=\varphi(t/n)^n=e^{-|t|}.
$$
Thus the <stability of the Cauchy distribution> implies that $S_n/n$ has the standard Cauchy distribution for every $n$. In particular,
$$
\frac{Y_1+\cdots+Y_n}{n}\xrightarrow{d}Z,
$$
where $Z$ is standard Cauchy and $\xrightarrow d$ denotes <convergence in distribution>.
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