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Let ${\{X_n}\}$ be a sequence of independent random variables and the stable distribution of order alpha $(0\le\alpha\le2)$.
Show that $$\sum_{k=1}^{n}\frac{X_k}{n^{\frac{1}{\alpha }}}$$ if ${X_n}$ is $X_k$s same distribution.
I can't find anywhere this theory. Can somebody help me?
This is only true if the location parameter is zero.
(In which case you can use the characteristic function of the stable distribution $e^{-|ct|^\alpha (1-i\beta sign(t) tan(\pi\alpha/2))}$ to proof it for $\alpha\neq 1$)
As a counter example consider the normal distribution (for which $\alpha=2$)
$$Y = \frac{1}{\sqrt{n}}\sum_{i=1}^n X_i$$
In this case you have $\sigma_Y = \sigma_X$ but $\mu_Y \neq \mu_X$ and instead $\mu_Y = \sqrt{n} \mu_X$. So $Y$ is not similarly distributed as $X$ (it is the same family though, maybe that is what you meant).
