Let $a$ be a real number. Let $m$ and $n$ be in $\mathbb{N}$.
a) Show that $S_{m+n} - S_{m}$ and $S_{n}$ have the same distribution.
b) Let $b$ be a real number. Show $P\left(S_{m+n} \geqslant (n+m)b\right) \geqslant P\left(S_{n} \geqslant nb\right) P\left(S_{m} \geqslant mb\right)$.

Assume that $X$ is constant equal to $a \in \mathbb{R}$. Show that $X$ is infinitely divisible.

Let $n$ be a non-zero natural number. We set $X_n = \sum_{k=1}^{n} \frac{\varepsilon_k}{2^k}$ where $(\varepsilon_n)_{n \geqslant 1}$ is a sequence of independent random variables taking values in $\{-1,1\}$ with $\mathbb{P}(\varepsilon_n = 1) = \mathbb{P}(\varepsilon_n = -1) = 1/2$.
Show that $X_n$ and $-X_n$ have the same distribution for all $n \in \mathbb{N}^{\star}$.

Let $n$ be a non-zero natural number. We set $$\forall n \in \mathbb{N}^{\star}, \quad X_n = \sum_{k=1}^{n} \frac{\varepsilon_k}{2^k}$$ where $(\varepsilon_n)_{n \geqslant 1}$ is a sequence of independent random variables taking values in $\{-1,1\}$ with $\mathbb{P}(\varepsilon_n = 1) = \mathbb{P}(\varepsilon_n = -1) = 1/2$ for all $n \geqslant 1$.
Show that $X_n$ and $-X_n$ have the same distribution for all $n \in \mathbb{N}^{\star}$.