Let $E$ be a Euclidean space of dimension $n \geqslant 1$ equipped with an orthonormal basis $(e_{1}, \ldots, e_{n})$. Let $\varepsilon_{1}, \ldots, \varepsilon_{n} : \Omega \rightarrow \{-1, 1\}$ be Rademacher random variables that are independent of each other. We set $X = \sum_{i=1}^{n} \varepsilon_{i} e_{i}$. We assume that $C$ is a closed convex set of $E$ that meets $X(\Omega)$ in a single vector $u$. Show that $\frac{1}{4} d(X, u)^{2}$ follows a binomial distribution with parameters $n$ and $1/2$.

Let $x \in \mathbb{R}$ such that $x > 1$. Show that we define the probability distribution of a random variable $X$ taking values in $\mathbb{N}^{*}$ by setting $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$

Let $X$ be a random variable that follows the zeta distribution with parameter $x > 1$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$ Give a necessary and sufficient condition on $x$ for $X$ to have a finite expectation. Express this expectation using $\zeta$.

Let $X$ be a random variable that follows the zeta distribution with parameter $x > 1$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$ For all $k \in \mathbb{N}$, give a necessary and sufficient condition on $x$ for $X^{k}$ to have a finite expectation. Express this expectation using $\zeta$.

Let $X$ be a random variable that follows the zeta distribution with parameter $x > 1$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$ Using the result of Q26, deduce the variance of $X$.

Let $X$ be a random variable that follows the zeta distribution with parameter $x > 1$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$ Show that, for all $a \in \mathbb{N}^{*}$, $$\mathbb{P}\left(X \in a\mathbb{N}^{*}\right) = \frac{1}{a^{x}}$$

Let $x$ be a real number such that $x > 1$ and let $X$ be a random variable that follows the zeta distribution with parameter $x$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$ For all $n \in \mathbb{N}^{*}$, denote by $B_{n}$ the event $B_{n} = \bigcap_{k=1}^{n} (X \notin p_{k}\mathbb{N}^{*})$, where $p_1 < p_2 < \cdots$ are the prime numbers in increasing order.
Show that $\lim_{n \rightarrow \infty} \mathbb{P}(B_{n}) = \mathbb{P}(X = 1)$. Deduce that $$\forall x \in {]1,+\infty[}, \quad \frac{1}{\zeta(x)} = \lim_{n \rightarrow +\infty} \prod_{k=1}^{n} \left(1 - \frac{1}{p_{k}^{x}}\right)$$

Let $U_{n}$ and $V_{n}$ be two independent random variables each following the uniform distribution on $\llbracket 1, n \rrbracket$. We denote by $W_{n} = U_{n} \wedge V_{n}$ (the GCD of $U_n$ and $V_n$).
For all $k \in \mathbb{N}^{*}$, show that $$\mathbb{P}\left(W_{n} \in k\mathbb{N}^{*}\right) = \left(\frac{\lfloor n/k \rfloor}{n}\right)^{2}$$

Let $U_{n}$ and $V_{n}$ be two independent random variables each following the uniform distribution on $\llbracket 1, n \rrbracket$, and $W_{n} = U_{n} \wedge V_{n}$. We admit that, for all $k \in \mathbb{N}^{*}$, the sequence $(\mathbb{P}(W_{n} = k))_{n \in \mathbb{N}^{*}}$ converges to a real number $\ell_{k}$.
Show that $$\forall \varepsilon > 0, \quad \exists M \in \mathbb{N}^{*} \text{ such that } \forall m \in \mathbb{N}^{*},\ m \geqslant M \Longrightarrow 1 - \varepsilon \leqslant \sum_{k=1}^{m} \ell_{k} \leqslant 1$$

Let $(\ell_k)_{k \in \mathbb{N}^*}$ be the limits defined in Q33, where $\ell_k = \lim_{n \to \infty} \mathbb{P}(W_n = k)$ and $W_n = U_n \wedge V_n$ for independent uniform random variables $U_n, V_n$ on $\llbracket 1, n \rrbracket$.
Using the result of Q33, deduce that $(\ell_{k})_{k \in \mathbb{N}^{*}}$ defines a probability distribution on $\mathbb{N}^{*}$.

Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables (taking values in $\{1,-1\}$ each with probability $1/2$). For every $n \in \mathbb{N}^{*}$, we set $Y_{n} = \frac{1}{2}\left(X_{n}+1\right)$ and $Z_{n} = \sum_{j=1}^{n} Y_{j}$. Determine the distribution of the random variable $Y_{n}$ and that of the random variable $Z_{n}$.

Let $W$ be a random variable on $\mathbb{N}^{*}$ that follows the probability distribution $(\ell_k)_{k \in \mathbb{N}^*}$, where $\ell_k = \lim_{n \to \infty} \mathbb{P}(W_n = k)$ and $W_n = U_n \wedge V_n$ for independent uniform random variables $U_n, V_n$ on $\llbracket 1, n \rrbracket$.
We admit that for all $B \subseteq \mathbb{N}^*$, $\mathbb{P}(W \in B) = \lim_{n \to \infty} \mathbb{P}(W_n \in B)$, and that if $X$ and $Y$ are two random variables taking values in $\mathbb{N}^*$ with $\mathbb{P}(X \in a\mathbb{N}^*) = \mathbb{P}(Y \in a\mathbb{N}^*)$ for all $a \in \mathbb{N}^*$, then $X$ and $Y$ have the same distribution.
Specify the distribution of $W$. By considering $\ell_{1}$, what can we then conclude?

Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables. We denote, for every integer $n \geqslant 1$, $S_{n} = \sum_{j=1}^{n} X_{j}$. Let $k$ be an integer such that $-n \leqslant k \leqslant n$. Show that, if $n$ and $k$ do not have the same parity, then $\mathbb{P}\left(S_{n} = k\right) = 0$.

Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables. We denote $S_{n} = \sum_{j=1}^{n} X_{j}$. Let $k$ be an integer such that $-n \leqslant k \leqslant n$. Show that, if $n$ and $k$ have the same parity, $\mathbb{P}\left(S_{n} = k\right) = \binom{n}{(k+n)/2} \frac{1}{2^{n}}$.

Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables and $S_{n} = \sum_{j=1}^{n} X_{j}$. For every $n \in \mathbb{N}^{*}$ and every $k \in \mathbb{Z}$, by setting $p_{n}(k) = \mathbb{P}\left(S_{n} = k\right)$, show that $$\frac{p_{n+1}(k) - p_{n}(k)}{\tau} = \frac{\delta^{2}}{2\tau} \frac{p_{n}(k+1) - 2p_{n}(k) + p_{n}(k-1)}{\delta^{2}}$$

Using the result of Q40, deduce a probabilistic interpretation of the stability condition studied in Part III (i.e., the condition on $r = \frac{\tau}{\delta^2}$ found in Q34).

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 $n \in \mathbb{N}^{\star}$.

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 $X_{n}$, where $g_{n}(t) = \sum_{k=0}^{+\infty} P(X_{n} = k) t^{k}$.
Determine the distributions of $X_{1}, X_{2}$ and $X_{3}$ and then the functions $g_{1}, g_{2}$ and $g_{3}$.

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.
Let $n$ and $k$ be two integers greater than or equal to 1. Establish that $$P(X_{n} = k) = \frac{k-1}{n+1} P(X_{n-1} = k-1) + \frac{n+1-k}{n+1} P(X_{n-1} = k).$$

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 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 $X_{n}$.
Prove that, for all integers $n \in \mathbb{N}^{*}$ and all real $t$, $$g_{n}(t) = \frac{1}{n+1} \sum_{k=1}^{n+1} t^{k}$$

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.
Identify the distribution of $X_{n}$ and give its expectation.

We consider a balanced urn with $a_{0} = 1, b_{0} = 0, a = d = 0$ and $b = c = 1$. In other words, the urn initially contains one white ball and, at each draw, we add a ball of the opposite color to the one that was drawn.
By listing all possible outcomes, give the distribution of $X_{3}$.

We consider a balanced urn with $a_{0} = 1, b_{0} = 0, a = d = 0$ and $b = c = 1$. In other words, the urn initially contains one white ball and, at each draw, we add a ball of the opposite color to the one that was drawn. For all real $u$ and $v$, we set $P_{0}(u,v) = u^{a_{0}} v^{b_{0}}$ and $P_{n}(u,v) = \sum_{\omega \in \Omega_{n}} u^{b(\omega)} v^{n(\omega)}$.
Verify that $P_{3}(u,v) = uv^{3} + 4u^{2}v^{2} + u^{3}v$.

We consider a general balanced urn with parameters $a_{0}, b_{0}, a, b, c, d \in \mathbb{N}$ satisfying $a + b = c + d = s$. For $n \geqslant 1$, $\Omega_{n}$ denotes the set of possible outcomes of $n$ draws, and $X_{n}$ denotes the number of white balls present in the urn after $n$ draws. For $\omega \in \Omega_{n}$, $b(\omega)$ denotes the number of white balls present in the urn at the end of the $n$ draws modeled by $\omega$.
Show that, for all $n \in \mathbb{N}^{*}$ and all $k \in \mathbb{N}$, $$P(X_{n} = k) = \frac{\operatorname{card}(\{\omega \in \Omega_{n} ; b(\omega) = k\})}{\operatorname{card}(\Omega_{n})}.$$