let $X\in\mathbb{R}^n$ be a random vector with i.i.d. entries $X_i=Y_i-\mu$, where the $X_i$ are distributed according to a Levy distribution with stability index $\mu\in(1,2)$ and arbitrary skewness parameter $\beta\in(-1,1)$. By the Levy distribution, I mean the distribution with characteristic function given by: $\log \hat{L}(k)=-C|k|^{\mu}(1+i \beta \operatorname{sign}(k) \tan (\pi \mu / 2))$ for some scale parameter $C>0$. Note that we have $E(X_i)=0$.

I now expect that $E(\|X\|_2)=E(\left( \sum_{i=1}^n X_i^2\right)^{1/2})=C n^{1/\mu}$ holds for some constant $C$, but I am not sure how to prove this. Does anyone have a proof of this? Or is my intuition wrong? This intuition comes from the fact that random matrices with i.i.d. entries coming from this distribution have an operator norm that scales like that, as shown in https://arxiv.org/pdf/cond-mat/0602087.pdf.