If $X_1,X_2,...$ are i.i.d. random variables such that $\phi (t)=E(e^{itX_{1}})=e^{-c|t|^{\alpha }}$, where $c>0$ and $\alpha\in(0,2]$, then show:
- For $1<\alpha\leq2$, show that $\frac{X_1+...+X_n}{n}$ converges to $0$ in probability.
- For $\alpha=1$, show that $\frac{X_1+...+X_n}{n}$ does not converge to a constant in probability.
- For $0<\alpha<1$, show that the distribution of $\frac{X_1+...+X_n}{n}$ does not converge weakly to any probability measure.
- For any $\alpha\in(0,2]$, show that there is some $\beta$ such that the distribution of $\frac{X_1+...+X_n}{n^{\beta}}$ does converge to some non-degenerate probability measure.