91Ó°ÊÓ

Let \(X_{1}, X_{2}, \ldots\) be i.i.d. random variables with density $$ f(x)=\frac{1}{|x|^{3}} \mathbb{1}_{\mathbb{R} \backslash[-1,1]}(x) $$ Then \(\mathbf{E}\left[X_{1}^{2}\right]=\infty\) but there are numbers \(A_{1}, A_{2}, \ldots\), such that $$ \frac{X_{1}+\ldots+X_{n}}{A_{n}} \stackrel{n \rightarrow \infty}{\Longrightarrow} \quad \mathcal{N}_{0,1} $$ Determine one such sequence \(\left(A_{n}\right)_{n \in \mathbb{N}}\) explicitly.

Let \(X\) be a real random variable with characteristic function \(\varphi\). \(X\) is called lattice distributed if there are \(a, d \in \mathbb{R}\) such that \(\mathbf{P}[X \in a+d \mathbb{Z}]=1\). Show that \(X\) is lattice distributed if and only if there exists a \(u \neq 0\) such that \(|\varphi(u)|=1\)