We assume $m>1$. We study the Galton-Watson process starting with $k$ individuals in generation 0, with $W_n$ the number of individuals in generation $n$. We define $u_n$ and $u_n^{(r)}$ as above.
Let $n\in\mathbb{N}^*$ and $r$ a natural integer greater than or equal to 2. Show the relation $$u_n^{(r)}=\sum_{i=1}^{n-1}u_i u_{n-i}^{(r-1)}$$

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 $k \in \mathbb{N}^{*}$, $$P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$$ For all $n \in \mathbb{N}^{*}$, we set $S_{n} = X_{1} + \cdots + X_{n}$, and $u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$.
Show, for all $n \in \mathbb{N}^{*}$, $$E\left(\left|S_{n}\right|\right) = \frac{2}{\pi} u_{n}$$ We will use the integral expression for the absolute value obtained in question I.A.5.

Let $Y$ be a random variable taking values in $\mathbb{N}$ almost surely, and which admits an expectation. Show that $$\mathbb{E}(Y) = \sum_{k=1}^{+\infty} \mathbb{P}(Y \geqslant k)$$

Let $X : \Omega \rightarrow \mathbb{R}$ be a real-valued random variable. We assume that there exist two strictly positive reals $a$ and $b$ such that, for all non-negative real $t$,
$$\mathbb{P}(|X| \geqslant t) \leqslant a \exp(-bt^{2})$$
Show that
$$\mathbb{E}(X^{2}) = 2 \int_{0}^{+\infty} t \mathbb{P}(|X| \geqslant t) \, dt$$
You may denote $X^{2}(\Omega) = \{y_{1}, \ldots, y_{n}\}$ with $0 \leqslant y_{1} < y_{2} < \cdots < y_{n}$.

Let $X : \Omega \rightarrow \mathbb{R}$ be a real-valued random variable. We assume that there exist two strictly positive reals $a$ and $b$ such that, for all non-negative reals $t$,
$$\mathbb{P}(|X| \geqslant t) \leqslant a \exp(-bt^{2})$$
Show that
$$\mathbb{E}(X^{2}) = 2 \int_{0}^{+\infty} t \mathbb{P}(|X| \geqslant t) \, dt$$
You may denote $X^{2}(\Omega) = \{y_{1}, \ldots, y_{n}\}$ with $0 \leqslant y_{1} < y_{2} < \cdots < y_{n}$.