Suppose $X_{1},X_{2}$... are independent, identically Linnik(${\alpha}$)-distributed random variables and that N$\epsilon Fs(p)$ (First Success) and that N and $X_{1},X_{2}$...are independent. Show that $p^{\frac{1}{\alpha }}(X_{1}+...+X_{N})$ is Linnik(${\alpha}$)-distributed..
My thought process so far is that we know the characteristic function of a Linnik(${\alpha}$)-distributed RV is $\frac{1}{1+|t|^{\alpha }}$ and that $\varphi _{N}(t) = g_{N}(\varphi _{X}(t))$ where $(\varphi _{X}(t))$ is the characteristic function of the Linnik-distribution. We also know that $g_{N}(t)$ = $\frac{pt}{1-qt}$ so plugging what we know in gives
$\varphi _{N}(t) = \frac{p\left(\frac{1}{1+|t|^{\alpha }}\right)}{1-q\left(\frac{1}{1+|t|^{\alpha }}\right)}$
The algebra leaves you with $p^{\frac{1}{\alpha }}\left(\frac{p}{p+|t|^{\alpha}}\right)$ so I am wondering how I can show from here that this is Linnik-distributed.
Are my methods so far correct, and what might be my next step?
Actually, it seems that you computed the characteristic function of $Y:=X_1+\dots+X_N$, which is indeed $$\mathbb E\left[e^{itY}\right]=\frac{p\left(\frac{1}{1+|t|^{\alpha }}\right)}{1-q\left(\frac{1}{1+|t|^{\alpha }}\right)}.$$ To get that of $p^{1/\alpha}Y$, we have to replace $t$ by $p^{1/\alpha}$ and this gives the wanted expression.