In this subsection, $X$ is a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$.
Show that there exists a unique real sequence $\left(\lambda_{i}\right)_{i \in \mathbb{N}^{*}}$ such that, for all $k \in \mathbb{N}^{*}$ $$k\mathbb{P}(X = k) = \sum_{j=1}^{k} j\lambda_{j} \mathbb{P}(X = k-j)$$

In this subsection, $X$ is a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$.
For all $k \in \mathbb{N}^{*}$, show $$\left|\lambda_{k}\right| \mathbb{P}(X = 0) \leqslant \mathbb{P}(X = k) + \sum_{j=1}^{k-1} \left|\lambda_{j}\right| \mathbb{P}(X = k-j) \leqslant (1 - \mathbb{P}(X = 0))\left(1 + \sum_{j=1}^{k-1} \left|\lambda_{j}\right|\right)$$

In this subsection, $X$ is a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$.
For all $k \in \mathbb{N}^{*}$, show: $1 + \sum_{j=1}^{k} \left|\lambda_{j}\right| \leqslant \frac{1}{\mathbb{P}(X = 0)^{k}}$.

In this subsection, $X$ is a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$.
Show that there exists a unique real sequence $\left(\lambda_{i}\right)_{i \in \mathbb{N}^{*}}$ such that, for all $k \in \mathbb{N}^{*}$ $$k\mathbb{P}(X = k) = \sum_{j=1}^{k} j\lambda_{j} \mathbb{P}(X = k-j)$$

In this subsection, $X$ is a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$, and $\left(\lambda_{i}\right)_{i \in \mathbb{N}^{*}}$ is the unique real sequence such that for all $k \in \mathbb{N}^{*}$, $k\mathbb{P}(X = k) = \sum_{j=1}^{k} j\lambda_{j} \mathbb{P}(X = k-j)$.
For all $k \in \mathbb{N}^{*}$, show $$\left|\lambda_{k}\right| \mathbb{P}(X = 0) \leqslant \mathbb{P}(X = k) + \sum_{j=1}^{k-1} \left|\lambda_{j}\right| \mathbb{P}(X = k-j) \leqslant (1 - \mathbb{P}(X = 0))\left(1 + \sum_{j=1}^{k-1} \left|\lambda_{j}\right|\right)$$

In this subsection, $X$ is a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$, and $\left(\lambda_{i}\right)_{i \in \mathbb{N}^{*}}$ is the unique real sequence such that for all $k \in \mathbb{N}^{*}$, $k\mathbb{P}(X = k) = \sum_{j=1}^{k} j\lambda_{j} \mathbb{P}(X = k-j)$.
For all $k \in \mathbb{N}^{*}$, show: $1 + \sum_{j=1}^{k} \left|\lambda_{j}\right| \leqslant \frac{1}{\mathbb{P}(X = 0)^{k}}$.

We have an infinite supply of black and white balls. An urn initially contains one black ball and one white ball. We perform a sequence of draws according to the following protocol:

• we randomly draw a ball from the urn;
• we replace the drawn ball in the urn;
• we add to the urn a ball of the same color as the drawn ball.

We define the sequence $(X_{n})_{n \in \mathbb{N}}$ of random variables by $X_{0} = 1$ and, for all integers $n \geqslant 1$, $X_{n}$ gives the number of white balls in the urn after $n$ draws. We denote by $g_{n}$ the generating function of the random variable $X_{n}$.
Using the recurrence relation from question 8, deduce that, for all integers $n$ greater than or equal to 1 and all real $t$, $$g_{n}(t) = \frac{t^{2} - t}{n+1} g_{n-1}^{\prime}(t) + g_{n-1}(t)$$

We consider the functions $F$ and $G$ defined by the formulas $$\begin{aligned} & \forall x \in ]-1,1[ , \quad F ( x ) = \sum _ { n = 0 } ^ { + \infty } P \left( S _ { n } = 0 _ { d } \right) x ^ { n } \\ & \forall x \in [ - 1,1 ] , \quad G ( x ) = \sum _ { n = 1 } ^ { + \infty } P ( R = n ) x ^ { n } \end{aligned}$$ If $k$ and $n$ are positive integers such that $k \leq n$, show that $$P \left( \left( S _ { n } = 0 _ { d } \right) \cap ( R = k ) \right) = P ( R = k ) P \left( S _ { n - k } = 0 _ { d } \right) .$$ Deduce that $$\forall n \in \mathbb{N}^{*} , \quad P \left( S _ { n } = 0 _ { d } \right) = \sum _ { k = 1 } ^ { n } P ( R = k ) P \left( S _ { n - k } = 0 _ { d } \right) .$$