Computing Char. Function of a Random Variable I'm trying to do the following problem:

Let $X_1, X_2, \ldots$ be an i.i.d. sequence of random variables such that for some $\alpha \in (0,2)$, 
$$ P(X_1 > x) = P(X_1 < -x), \quad P(|X_1|>x) =x^{-\alpha}, \quad x \geq 1.$$
Show that $S_n/n^{1/\alpha}$ converges weakly to variable with characteristic function $\phi(t) = \exp(-C|t|^\alpha)$ for some $C>0$. 

Here $S_n = X_1 + \ldots + X_n$. I know that by the convergence theorems for characteristic functions, it suffices to show that the characteristic functions of $S_n/n^{1/\alpha}$ converge to $\phi(t) = \exp(-C|t|^\alpha)$. 
If $X_i$ has characteristic function $\varphi(t)$, then the characteristic function of $S_n/n^{1/\alpha}$ is $[\varphi(t/n^{1/\alpha})]^n$.
I don't know how to get the characteristic function of $X_i$ though. The information given tells me $X_i$ is symmetric and $P(X_i \in [-1,1]) = 0$. 
Any advice on how to proceed? Thanks. 
Edit: I think I've found the probability density function for $X_i$ to be $f(t) = \begin{cases} \alpha |x|^{-\alpha - 1}/2 & x\notin (-1,1) \\ 0 &x \in (-1,1) \end{cases}$.
However, if I put that into the definition of the characteristic function and try to compute it, I get this far $$\varphi(t) = \int_{1}^\infty \cos(tx)x^{-\alpha} dx $$
which is difficult (impossible?) to compute for arbitrary $\alpha$. 
 A: The integral defining the characteristic function reads:
$$ \begin{eqnarray}
  \varphi_X(t) &=& \int_1^\infty \alpha \cos(t x) x^{-\alpha-1} \mathrm{d} x = 1 + \alpha  \int_1^\infty \left(\cos(t x)-1\right) x^{-\alpha-1} \mathrm{d} x \\&=& 1 - 2 \alpha  \int_1^\infty \sin^2 \left(\frac{t x}{2}\right) x^{-\alpha-1} \mathrm{d} x \stackrel{u = |t| x}{=} 1-2 \alpha |t|^\alpha \underbrace{\int_{|t|}^\infty \sin^2\left(\frac{u}{2}\right) u^{-\alpha-1} \mathrm{d}u}_{\mathcal{I}_\alpha(|t|)}
\end{eqnarray} 
$$
The remaining integral $\mathcal{I}_\alpha(|t|)$ is well behaved for small $|t|$. Thus
$$
   \left(\varphi_X\left(\frac{t}{n^{1/\alpha}}\right)\right)^n = \left(1-\frac{2}{n} \alpha \mathcal{I}_\alpha(0) |t|^\alpha + \mathcal{o}\left(\frac{1}{n}\right) \right)^n \rightarrow \exp\left(-|t|^\alpha \left(2 \alpha \mathcal{I}_\alpha(0) \right) \right)
$$
The integral $\mathcal{I}_\alpha(0)$ can be evaluated in closed form:
$$
 2 \alpha \mathcal{I}_\alpha(0) = 2 \alpha \int_0^\infty \sin^2\left(\frac{u}{2}\right) \frac{\mathrm{d}u}{u^{\alpha+1}} = \frac{\pi/2}{\sin\left(\frac{\pi \alpha}{2}\right)} \frac{1}{\Gamma\left(\alpha\right)}
$$
but it does not seem required by your problem, so I am skipping it.
