For $p\in(0,2)$ let $X$ be a symmetric r.v. with distribution $\P(|X| > x)=\min\{1,x^{-p}\}$. If $X_1,\ldots,X_n$ denote independent copies of $X$ then the r.v.
$$
S_n\colon=\frac1{n^p}\sum_{j=1}^nX_j
$$
converge in distribution to a symmetric $p$-stable r.v. with parameter
$$
\s_p^p=2\int_0^\infty\frac{1-\cos(u)}{u^{1+p}}\,dx~.
$$
The characteristic function $\vp_X(t)$ of $X$ is given by
$$
1-\p_X(t)
=p\int_1^\infty\frac{1-\cos(tx)}{x^{1+p}}\,dx
=pt^p\int_t^\infty\frac{1-\cos(u)}{u^{1+p}}\,du~.
$$
As $t\to0$ we get $\vp_X(t)\sim1-C_p t^p$ where
$$
C_p=p\int_0^\infty\frac{1-\cos(u)}{u^{1+p}}\,dx~.
$$
Hence we get for the characteristic function $\vp_n$ of $S_n$:
\begin{eqnarray*}
\vp_n(t)
&=&\E\exp(itS_n)
=(\E\exp(itn^{-1/p}X)^n\\
&=&\vp_X(tn^{-1/p})
=(1-(1-\vp_X(tn^{-1/p})))^n
\stackrel{n\to\infty}{\longrar}\exp(-C_p|t|^p)~.
\end{eqnarray*}