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Defining Levy distribution with log charactersitic function $-a |\xi|^\alpha$ for $0 <\alpha< 2$ gives large $x$ asymptotic
$l_\alpha(a,x) = \frac{1}{\pi |x|^{\alpha+1}} \sum \frac{a^m \sin(\frac{\alpha m \pi}{2})\Gamma(am+1)^m}{m! |x|^m}$
Width of the Levy distribution after $N$ steps is $N^{1/\alpha}$.  These are covered in standard courses such as this one.