We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$. We also consider a sequence $\left(a_{n}\right)_{n \in \mathbb{N}^{*}}$ of non-negative real numbers. For all $n \in \mathbb{N}^{*}$, we set $T_{n} = \sum_{k=1}^{n} a_{k} X_{k}$. Show that if the series $\sum a_{n}^{2}$ is convergent, then the sequence $\left(E\left(\left|T_{n}\right|\right)\right)_{n \in \mathbb{N}^{*}}$ is convergent.
We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$. We also consider a sequence $\left(a_{n}\right)_{n \in \mathbb{N}^{*}}$ of non-negative real numbers. For all $n \in \mathbb{N}^{*}$, we set $T_{n} = \sum_{k=1}^{n} a_{k} X_{k}$.
Show that if the series $\sum a_{n}^{2}$ is convergent, then the sequence $\left(E\left(\left|T_{n}\right|\right)\right)_{n \in \mathbb{N}^{*}}$ is convergent.