Consider the following characteristic function and show that the distribution is infinitely divisible $$\phi(u)=exp(imu-\sigma\vert u\vert[1+i\beta\frac{2}{\pi}\text{sgn}(u)\text{log}\vert u\vert])$$ with $\sigma > 0, -1\le \beta\le 1$ and $m\in\mathbb{R}$.

My attempt:

So, first thing that sticks out is that this is the CF of an $\alpha$-stable distribution with $\alpha=1$. For example, if $\beta=0$, we have a Cauchy$(m,\sigma)$ distribution. Now consider the following $$(\phi_{\frac{1}{n}}(u))^n=\{[\phi(u)=exp(imu-\sigma\vert u\vert[1+i\beta\frac{2}{\pi}\text{sgn}(u)\text{log}\vert u\vert])]^\frac{1}{n}\}^n$$ Now, here's where I'm a little stuck, mostly just with how to write it/explain it from here. For example, if we consider $\beta=0$, the term in braces can just be written as the characteristic function of a Cauchy$(\frac{m}{n},\frac{\sigma}{n})$ distribution, and we can thus say it is infinitely divisible. Can I just make some generalization here of the form $$(\phi_{\frac{1}{n}}(u))^n=\{[\phi(u)=exp(i\frac{m}{n}u-\frac{\sigma}{n}\vert u\vert[1+i\beta\frac{2}{\pi}\text{sgn}(u)\text{log}\vert u\vert])]\}^n$$ and conclude that the term in braces is just the characteristic function of some $\alpha=1$-stable distribution with parameters $\frac{m}{n},\frac{\sigma}{n},\beta$, infinitely divisible as all $\alpha$-stable distributions are?


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