Let $X_{1}, X_{2}, \ldots, X_{n}$ be the sequence of i.i.d random variables with heavvy tailed distributions, i.e. $$p(x_{i}) \sim \frac{A}{x_{i}^{\alpha}}$$ as $x_{i} \rightarrow \infty$, where $p(x_{i})$ stands for the density of $X_{i}$.
The question is: how to estimate the asymtotic of the density $p(y)$, where $$Y = X_{1} + X_{2} + \ldots + X_{n}$$ ?
A pretty straightforward approach is the following: calculate the characteristic function of $X_{i}$, since the random variables are i.i.d, then $$\varphi_{Y}(t) = \varphi_{X_{1} + X_{2} + \ldots + X_{n}}(t) = (\varphi_{X_{1}}(t))^{n}$$ then apply the inverse Fourier transform to figure out the distribution of sum.
Are there any, say, 'elegant' ways to approach the problem above?