Consider a set of N positive random variables $x_i$ , i.i.d., with infinite variance. The distribution of each variable, decays asymptotically as a power law:
$P_{x_i}(x) \sim x^{-1-\alpha}$ for $x \Rightarrow \infty$, with $1<\alpha<2$
Consider now the sum of this set of variables; this is also, of course, a random variable. We can look for an expansion for its probability distribution with respect to the scaling parameter N (the size of the set involved in the sum). The leading term with the first correction are (scale as):
$\sum_{i=1}^N x_i = N \overline{x} + N^{\frac{1}{\alpha}}Y$, where $Y$ is a random variable and $\overline{x} = \int x' P_{x_i}(x') dx'$.
What is the next term in this expansion? How does it scale with N?
In essence I'm looking for an extended version of Edgeworth expansion for the case of infinite variance random variables to obtain the scaling behaviour (with N) of the next subleading terms.
Note that now there is an additional difficulty
to overcome with respect to the finite variance Edgeworth expansion. In that case we perform a perturbation of the higher cumulant terms, keppeng the gaussian form and expanding all the rest that appear in exponent.
Now the cumulant are not defined anymore (the variance diverge).
It is also possible to find an expansion form considering an asimptically power law distribution for the variable $x_i$ but with finite variance:
https://arxiv.org/abs/1103.4306
What I want instead is a similar expansion for the infinite variance random variables case.