In the formula $f(x) = \sum_{i=0}^{+\infty} q_i g\left(x - y_i\right)$, show that the convergence of the series is normal on every segment of $\mathbb{R}$. One may use question 3c.

Suppose that $g$ is continuous. Show that $f$ is uniformly continuous.

Suppose that $g$ is of class $\mathscr{C}^1$. Show that $g'$ is bounded. Deduce that $f$ is of class $\mathscr{C}^1$, that $f'$ is bounded and uniformly continuous and that for all $x \in \mathbb{R}$, $$f'(x) = g'(x) + \sum_{i=0}^{+\infty} p_i f'\left(x - x_i\right)$$

Throughout this question, $\mathcal{S}$ is a simplex in $\mathbb{R}^n$ such that $0 \in \mathring{\mathcal{S}}$. We want to show that $$\operatorname{Card}\left(\mathring{\mathcal{S}} \cap \mathbb{Z}^n\right) \geqslant 2\left\lfloor \operatorname{Vol}(\mathcal{S})\left(\frac{a(\mathcal{S})}{a(\mathcal{S})+1}\right)^n \right\rfloor + 1$$ We then set $a = a(\mathcal{S})$, and $k = \rfloor \operatorname{Vol}(\mathcal{S})\left(\frac{a}{a+1}\right)^n \lfloor$.
10a. Express, for $\beta \in \mathbb{R}^*$ and $x \in \mathbb{R}^n$, $\operatorname{Vol}(\beta \mathcal{S})$ and $\operatorname{Vol}(\mathcal{S} - x)$. Show that for $\lambda \in [0,1[$ sufficiently close to $1$, $\operatorname{Vol}\left(\frac{\lambda a}{a+1}\mathcal{S}\right) > k$.
10b. For $\lambda$ as in the previous question, let $v_0, \ldots, v_k$ be the $k+1$ distinct points in $\frac{\lambda a}{a+1}\mathcal{S}$ satisfying $v_i - v_j \in \mathbb{Z}^n$ for all $i, j$, whose existence is guaranteed by Theorem 1. Show that the points $v_i - v_j$ are in $\lambda \mathcal{S}$. Deduce that the $v_i - v_j$ are in $\mathring{\mathcal{S}}$.
10c. Show that there exists an index $j \in \{0, \ldots, k\}$ such that the $(2k+1)$ points $0, \pm(v_i - v_j)$, for $i \in \{0, \ldots, k\} \setminus \{j\}$ are distinct. Deduce the statement of question 10, then that $$\operatorname{Card}\left(\mathring{\mathcal{S}} \cap \mathbb{Z}^n\right) \geqslant \operatorname{Vol}(\mathcal{S})\left(\frac{a(\mathcal{S})}{2}\right)^n$$

Let $f \in \mathbb{R}^{N}$ and $J_{f} : \Sigma_{N} \rightarrow \mathbb{R}$ defined by $J_{f}(p) = H_{N}(p) + \sum_{i=1}^{N} p_{i} f_{i}$. We denote $J_{f,*} = \sup\{J_{f}(p) \mid p \in \Sigma_{N}\}$ and $\Sigma_{N}(f) = \{p \in \Sigma_{N} \mid J_{f}(p) = J_{f,*}\}$.
Identify $\Sigma_{N}(f)$. Show that $J_{f,*} = \ln\left(\sum_{i=1}^{N} e^{f_{i}}\right)$.

Let $f \in \mathbb{R}^{N}$ and $J_{f,*} = \ln\left(\sum_{i=1}^{N} e^{f_{i}}\right)$. We consider $F : ]0, +\infty[ \rightarrow \mathbb{R}$ the function defined by $F(\beta) = \frac{1}{\beta} \ln\left(\sum_{i=1}^{N} e^{\beta f_{i}}\right)$.
Show that $F$ is differentiable and calculate its derivative $F'$. Show further that for all $\beta \in ]0, +\infty[$, there exists $p(\beta) \in \Sigma_{N}(\beta f)$ such that $F'(\beta) = -\frac{1}{\beta^{2}} H_{N}(p(\beta))$.

We consider $F : ]0, +\infty[ \rightarrow \mathbb{R}$ the function defined by $F(\beta) = \frac{1}{\beta} \ln\left(\sum_{i=1}^{N} e^{\beta f_{i}}\right)$ where $f \in \mathbb{R}^{N}$.
Study the limits of $F$ at 0 and at $+\infty$.

Two simplexes $\mathcal{S}$ and $\mathcal{S}'$ in $\mathbb{R}^n$ are called equivalent if there exist an enumeration of the vertices $s_0, s_1, \ldots, s_n$ of $\mathcal{S}$, and $s_0', s_1', \ldots, s_n'$ of $\mathcal{S}'$, and a matrix $A$ in $\mathrm{GL}_n(\mathbb{Z})$ such that $A(s_i - s_0) = s_i' - s_0'$ for all $i = 1, \ldots, n$.
Show that an integer simplex $\mathcal{S}$ is equivalent to an integer simplex contained in the cube $[0, n!\operatorname{Vol}(\mathcal{S})]^n$.
One may use question 6 for a suitably chosen matrix $M$.

Let $(\Omega, \mathscr{A}, \mathbf{P})$ be a probability space and $X : \Omega \rightarrow \{1, \ldots, N\}$ a random variable with distribution $q \in \Sigma_{N}$. Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$ for $(i,j) \in \{1, \ldots, N\} \times \{1, \ldots, d\}$, $p \in \Sigma_{N}$ and $m \in \mathbb{R}^{d}$. We denote by $A \in \mathscr{M}_{d}(\mathbb{R})$ the square matrix of size $d \times d$ defined for all $(k,l) \in \{1, \ldots, d\}^{2}$ by $$A_{lk} = \sum_{i=1}^{N} p_{i}(M_{il} - m_{l})(M_{ik} - m_{k}).$$
Verify that if $Y : \Omega \rightarrow \{1, \ldots, N\}$ is a random variable with distribution $p$, then $A_{lk} = \mathbf{E}((g_{l}(Y) - m_{l})(g_{k}(Y) - m_{k}))$ and then that $A$ is a symmetric matrix such that $\theta^{T} A \theta \geqslant 0$ for all $\theta \in \mathbb{R}^{d}$.

Two simplexes $\mathcal{S}$ and $\mathcal{S}'$ in $\mathbb{R}^n$ are called equivalent if there exist an enumeration of the vertices $s_0, s_1, \ldots, s_n$ of $\mathcal{S}$, and $s_0', s_1', \ldots, s_n'$ of $\mathcal{S}'$, and a matrix $A$ in $\mathrm{GL}_n(\mathbb{Z})$ such that $A(s_i - s_0) = s_i' - s_0'$ for all $i = 1, \ldots, n$.
Theorem 2 states: For every strictly positive integer $k$, there exists a strictly positive constant $C(n,k)$ such that for every integer simplex $\mathcal{S}$ in $\mathbb{R}^n$ having exactly $k$ interior integer points, $\operatorname{Vol}(\mathcal{S}) \leqslant C(n,k)$.
Deduce from Theorem 2 that for every strictly positive integer $k$, there exist up to equivalence only finitely many integer simplexes in $\mathbb{R}^n$ having exactly $k$ interior points.

Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$, $p \in \Sigma_{N}$, $m \in \mathbb{R}^{d}$, and $A \in \mathscr{M}_{d}(\mathbb{R})$ defined by $A_{lk} = \sum_{i=1}^{N} p_{i}(M_{il} - m_{l})(M_{ik} - m_{k})$. We denote by $\widetilde{M} = (M \mid \mathbf{1}) \in \mathscr{M}_{N,d+1}(\mathbb{R})$ the augmented matrix obtained by adding a column of 1s to the right of $M$.
Let $\theta \in \mathbb{R}^{d}$ such that $\theta^{T} A \theta = 0$. We assume that $p_{i} \neq 0$ for all $1 \leqslant i \leqslant N$.
(a) Show that there exists $c \in \mathbb{R}$, which you will specify, such that for all $i \in \{1, \ldots, N\}$, we have $\sum_{l=1}^{d} M_{il} \theta_{l} = c$.
(b) Show that if $\ker \widetilde{M} = \{0\}$ then $\theta = 0$.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that $g$ is of class $\mathscr{C}^1$, with support in $[0, K]$ with $K > 0$. We admit that there exists a subsequence $\left(t_k\right)_{k \geqslant 0}$ of $\left(y_n\right)_{n \geqslant 0}$ such that the sequence of functions $\left(\xi_k\right)_{k \geqslant 0}$ defined by $$\xi_k : \mathbb{R} \rightarrow \mathbb{R}, \quad t \mapsto \xi_k(t) = f'\left(t + t_k\right)$$ converges uniformly on every segment of $\mathbb{R}$ to a function denoted $\xi$. Show that $\xi$ is constant, equal to $c$.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that $g$ is of class $\mathscr{C}^1$, with support in $[0, K]$ with $K > 0$. With $\xi$ constant equal to $c$ as shown in question 14c, conclude that $c = 0$.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that $g$ is of class $\mathscr{C}^1$, with support in $[0, K]$ with $K > 0$. We admit that $\lim_{x \rightarrow +\infty} \inf_{t \geqslant x} f'(t) = 0$. Deduce that $f'(t) \rightarrow 0$ when $t \rightarrow +\infty$.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that $g$ is of class $\mathscr{C}^1$, with support in $[0, K]$ with $K > 0$. Using the result that $f'(t) \rightarrow 0$ when $t \rightarrow +\infty$, show that for all $\ell \geqslant 0$, $f(t+\ell) - f(t) \rightarrow 0$ when $t \rightarrow +\infty$.

Let $\mathcal{S}$ be a simplex of $\mathbb{R}^n$ and $k$ an integer such that $\operatorname{Vol}(\mathcal{S}) > k$.
15a. Show that there exist $x \in [0,1[^n$ and $(k+1)$ elements of $\mathbb{Z}^n$ $u_0, \ldots, u_k$ such that $x \in \mathcal{S} - u_i$ for $i = 0, \ldots, k$. One may study the sets $(u + [0,1[^n) \cap \mathcal{S}$ when $u$ ranges over $\mathbb{Z}^n$; and admit — outside the CPGE curriculum — that the volume of a simplex is its Lebesgue measure, which is sub-additive.
15b. Deduce from this the existence of the $(k+1)$ points $v_0, \ldots, v_k$ that satisfy the conditions of Theorem 1.
15c. Prove Theorem 1, that is, here we assume only that $\operatorname{Vol}(\mathcal{S}) \geqslant k$.

Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$. For all $\theta \in \mathbb{R}^{d}$, let $f(\theta) = M\theta \in \mathbb{R}^{N}$, $Z(\theta) = \sum_{i=1}^{N} e^{f_{i}(\theta)}$ and $$p(\theta) = \left(\frac{e^{f_{1}(\theta)}}{Z(\theta)}, \ldots, \frac{e^{f_{N}(\theta)}}{Z(\theta)}\right) \in \Sigma_{N}$$ where $f(\theta) = (f_{1}(\theta), \ldots, f_{N}(\theta))$. The function $L : \mathbb{R}^{d} \rightarrow \mathbb{R}$ is defined by $$L(\theta) = \ln(Z(\theta)) - q^{T} M\theta.$$
Show that $L$ is of class $\mathscr{C}^{1}$ and calculate its gradient.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that only finitely many $p_i$ are strictly positive. We set $$g_0(x) = \begin{cases} \mathbb{P}(X > x) & \text{if } x \geqslant 0 \\ 0 & \text{if } x < 0 \end{cases}$$ For all $g \in \mathscr{F}$ (the set of positive bounded functions with support in a segment of $\mathbb{R}^+$), we denote by $Lg$ the unique solution of (E) bounded with support in $\mathbb{R}^+$. A sequence $\left(t_k\right)_{k \geqslant 0}$ satisfies property $(\mathscr{P})$ if $t_k \rightarrow +\infty$ and there exists a continuous bounded function $\mu : \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for every piecewise continuous $g \in \mathscr{F}$, $$Lg\left(t_k\right) \rightarrow \int_0^{+\infty} g(t)\mu(t)\,dt \quad \text{when} \quad k \rightarrow +\infty$$ Show, using question 14f, that for all $g \in \mathscr{F} \cap \mathscr{C}^1(\mathbb{R}, \mathbb{R}^+)$ and $\ell \geqslant 0$, $$\int_0^{+\infty} g(t)(\mu(t+\ell) - \mu(t))\,dt = 0$$

Under the same assumptions as question 15a, deduce that $\mu$ is constant.

Let $t_1, \ldots, t_n$ be strictly positive real numbers such that $\sum_{i=1}^n t_i = 1$ and let $N \geqslant n$ be an integer. We wish to show that there exist non-negative integers $p_1, \ldots, p_n$ and $q$ such that
i) $1 \leqslant q \leqslant N^{n-1}$,
ii) $\sum_{i=1}^n p_i = q$,
iii) $\left|qt_1 - p_1\right| \leqslant \frac{n}{N}$,
iv) for all $i = 2, \ldots, n$, $\left|qt_i - p_i\right| \leqslant \frac{1}{N}$.
16a. By considering the vectors with coordinates $\left(\{kt_2\}, \ldots, \{kt_n\}\right) \in [0,1[^{n-1}$ when $k$ ranges over $\{0, \ldots, N^{n-1}\}$, show that there exist integers $p_2, \ldots, p_n, q \geqslant 0$ satisfying conditions i) and iv).
16b. Conclude.

Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$, $q \in \Sigma_{N}$, and for all $\theta \in \mathbb{R}^{d}$, $f(\theta) = M\theta$, $Z(\theta) = \sum_{i=1}^{N} e^{f_{i}(\theta)}$, $$p(\theta) = \left(\frac{e^{f_{1}(\theta)}}{Z(\theta)}, \ldots, \frac{e^{f_{N}(\theta)}}{Z(\theta)}\right) \in \Sigma_{N},$$ and $L(\theta) = \ln(Z(\theta)) - q^{T} M\theta$. We denote $$\Sigma_{N}(\bar{g}, g) = \left\{p \in \Sigma_{N} \mid \sum_{i=1}^{N} p_{i} g_{k}(i) = \bar{g}_{k}, 1 \leqslant k \leqslant d\right\}.$$
Show that if $\theta$ is a critical point of $L$ (that is, a point where the gradient of $L$ vanishes) then $M^{T} p(\theta) = M^{T} q$ and $p(\theta) \in \Sigma_{N}(\bar{g}, g)$.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that only finitely many $p_i$ are strictly positive. We set $$g_0(x) = \begin{cases} \mathbb{P}(X > x) & \text{if } x \geqslant 0 \\ 0 & \text{if } x < 0 \end{cases}$$ and $Lg_0$ denotes the unique bounded solution of (E) with support in $\mathbb{R}^+$ for $g = g_0$. Show that $Lg_0(x) = 1$ for $x \geqslant 0$ and $Lg_0(x) = 0$ for $x < 0$.

Under the same assumptions as question 16a, and using the fact that $\mu$ is constant (question 15b) and that $\int_0^{+\infty} g_0(t)\,dt = \mathbb{E}(X)$, deduce that $\mu(t) = \dfrac{1}{\mathbb{E}(X)}$ for all $t \geqslant 0$.

The purpose of this question is to show that for any strictly positive integers $n$ and $k$, there exists a constant $\alpha(k,n) \in ]0,1[$ such that, if $t_1, \ldots, t_n$ are strictly positive real numbers satisfying $1 > \sum_{i=1}^n t_i > 1 - \alpha(k,n)$, then there exist non-negative integers $p_1, \ldots, p_n \geqslant 0$ and $q$ such that $$\sum_{i=1}^n p_i = q > 0, \quad \text{and for all } i = 1, \ldots, n, \quad (kq+1)t_i > kp_i.$$ We proceed by induction on $n$.
17a. Handle the case $n = 1$ by showing that the constant $\alpha(k,1) = \frac{1}{k+1}$ works.
We assume the statement is true up to rank $n-1 \geqslant 1$. In particular, $\alpha(k,n-1) > 0$ is defined for all $k \geqslant 1$. We set for $k \geqslant 1$ $$\alpha(k,n) = \frac{1}{4kN^{n-1}} \quad \text{where} \quad N = 1 + \max\left(\frac{4k}{\alpha(k,n-1)}, 2kn(n+1)\right).$$ We are given $t_1 \geqslant t_2 \geqslant \cdots \geqslant t_n > 0$, and we assume that $\sum_{i=1}^n t_i = 1 - \alpha$ with $0 < \alpha < \alpha(k,n)$.
17b. If $t_n < \alpha(k,n-1) - \alpha$, establish the statement at rank $n$.
17c. If $t_n \geqslant \alpha(k,n-1) - \alpha$, apply the result of question 16 to the $\frac{t_i}{1-\alpha}$, $i = 1, \ldots, n$. With its notation, show that $$\alpha(k,n) < \min\left(\frac{1}{n+1}, \frac{1}{2}\alpha(k,n-1)\right) \quad \text{and} \quad 1 - qk\frac{\alpha}{1-\alpha} \geqslant \frac{1}{2}.$$ Conclude by distinguishing the cases $i \geqslant 2$ and $i = 1$.

Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$, $q \in \Sigma_{N}$, and for all $\theta \in \mathbb{R}^{d}$, $f(\theta) = M\theta$, $Z(\theta) = \sum_{i=1}^{N} e^{f_{i}(\theta)}$, $$p(\theta) = \left(\frac{e^{f_{1}(\theta)}}{Z(\theta)}, \ldots, \frac{e^{f_{N}(\theta)}}{Z(\theta)}\right) \in \Sigma_{N},$$ and $L(\theta) = \ln(Z(\theta)) - q^{T} M\theta$.
Show that $L$ is of class $\mathscr{C}^{2}$ and that for all integers $1 \leqslant l, k \leqslant d$ we have $$\frac{\partial^{2} L}{\partial \theta_{l} \partial \theta_{k}}(\theta) = \sum_{i=1}^{N} p_{i}(\theta)(M_{il} - m_{l}(\theta))(M_{ik} - m_{k}(\theta))$$ where $m(\theta) = M^{T} p(\theta)$.