grandes-ecoles

Papers (191)
2025
centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38
2024
centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26
2023
centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15
2022
centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23
2021
centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44
2020
centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18
2019
centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26
2018
centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24
2017
centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26
2016
centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20
2015
centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24
2013
centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18
2011
centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10
2010
centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27
2016 x-ens-maths2__mp

46 maths questions

Q1a Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables View
Justify that for all $\ell \geqslant 0$ and $n \in \mathbb{N}$, $(N(0,\ell) = n+1) = (S_n \leqslant \ell < S_{n+1})$ up to a negligible set. Deduce that, up to negligible sets, $$\left(S_n \leqslant \ell\right) = (N(0,\ell) \geqslant n+1) \quad \text{and} \quad \left(S_n \geqslant \ell\right) \subset (N(0,\ell) \leqslant n+1).$$
Q1b Central limit theorem View
Suppose in this question that $X$ additionally admits a finite variance $V$. Show then that $$\forall \varepsilon > 0, \forall n \geqslant 1, \quad \mathbb{P}\left(S_n \leqslant n(m-\varepsilon)\right) \leqslant \frac{V}{\varepsilon^2 n}.$$
Q2 Discrete Random Variables Integral or Series Representation of Moments View
Let $Y$ be a random variable taking values in $\mathbb{N}$ almost surely, and which admits an expectation. Show that $$\mathbb{E}(Y) = \sum_{k=1}^{+\infty} \mathbb{P}(Y \geqslant k)$$
Q3a Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables View
Show that for all $n \in \mathbb{N}$ and $\ell \geqslant 0$, $$\mathbb{P}\left(S_n \leqslant \ell\right) \leqslant \mathbb{E}\left(\exp\left(\ell - S_n\right)\right)$$ then that $$\mathbb{P}\left(S_n \leqslant \ell\right) \leqslant e^{\ell} \mathbb{E}(\exp(-X))^n$$
Q3b Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables View
Deduce that $\mathbb{P}\left(S_n \leqslant \ell\right)$ tends to 0 when $n \rightarrow +\infty$ and that $$\mathbb{E}(N(0,\ell)) \leqslant \frac{e^{\ell}}{1 - \mathbb{E}(\exp(-X))}$$
Q3c Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables View
Show that for all $x \in \mathbb{R}, \ell \geqslant 0, k \in \mathbb{N}^*$ and $n \in \mathbb{N}^*$, $$\mathbb{P}\left(S_{n-1} < x \leqslant S_n, N(x, x+\ell) \geqslant k\right) \leqslant \mathbb{P}\left(S_{n-1} < x \leqslant S_n\right) \mathbb{P}(N(0,\ell) \geqslant k)$$ then that $$\mathbb{E}(N(x, x+\ell)) \leqslant \frac{e^{\ell}}{1 - \mathbb{E}(\exp(-X))}$$
Q4a Discrete Random Variables Monotonicity and Convergence of Sequences Defined via Expectations View
Let $K > 0$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ be a positive bounded function with support in $[0, K]$. The sequence of functions $f_n : \mathbb{R} \rightarrow \mathbb{R}$ is defined for $n \geqslant 0$ by $$f_n(x) = \sum_{k=0}^{n} \mathbb{E}\left(g\left(x - S_k\right)\right)$$ Show that for all $x \in \mathbb{R}$, the sequence $\left(f_n(x)\right)_{n \geqslant 0}$ is increasing. We denote by $f(x)$ its limit in $\mathbb{R} \cup \{+\infty\}$.
Q4b Discrete Random Variables Expectation of a Function of a Discrete Random Variable View
Let $K > 0$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ be a positive bounded function with support in $[0, K]$. The sequence of functions $f_n : \mathbb{R} \rightarrow \mathbb{R}$ is defined for $n \geqslant 0$ by $$f_n(x) = \sum_{k=0}^{n} \mathbb{E}\left(g\left(x - S_k\right)\right)$$ Show that if $g = \mathbb{1}_{[0,K]}$, then $f(x) = \mathbb{E}(N(x-K, x))$.
Q4c Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables View
Let $K > 0$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ be a positive bounded function with support in $[0, K]$. The sequence of functions $f_n : \mathbb{R} \rightarrow \mathbb{R}$ is defined for $n \geqslant 0$ by $$f_n(x) = \sum_{k=0}^{n} \mathbb{E}\left(g\left(x - S_k\right)\right)$$ Deduce that for all $x \in \mathbb{R}$ and $n \in \mathbb{N}$, $$0 \leqslant f_n(x) \leqslant \|g\|_{\infty} \frac{e^K}{1 - \mathbb{E}(\exp(-X))}$$
Q4d Discrete Random Variables Monotonicity and Convergence of Sequences Defined via Expectations View
Let $K > 0$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ be a positive bounded function with support in $[0, K]$. The sequence of functions $f_n : \mathbb{R} \rightarrow \mathbb{R}$ is defined for $n \geqslant 0$ by $$f_n(x) = \sum_{k=0}^{n} \mathbb{E}\left(g\left(x - S_k\right)\right)$$ Conclude that the sequence of functions $f_n$ converges pointwise to a positive bounded function $f$ whose support is included in $\mathbb{R}^+$.
Q5 Discrete Random Variables Expectation of a Function of a Discrete Random Variable View
Let $Y$ be a discrete random variable, independent of $X$, and $\varphi : \mathbb{R}^2 \rightarrow \mathbb{R}$ be a bounded function. Show that $$\mathbb{E}(\varphi(X, Y)) = \sum_{i=0}^{+\infty} p_i \mathbb{E}\left(\varphi\left(x_i, Y\right)\right)$$
Q6a Discrete Random Variables Expectation and Variance of Sums of Independent Variables View
Let $K > 0$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ be a positive bounded function with support in $[0, K]$. The sequence of functions $f_n : \mathbb{R} \rightarrow \mathbb{R}$ is defined for $n \geqslant 0$ by $$f_n(x) = \sum_{k=0}^{n} \mathbb{E}\left(g\left(x - S_k\right)\right)$$ Show that for all $n \in \mathbb{N}$ and $x \in \mathbb{R}$, $$f_{n+1}(x) = g(x) + \sum_{i=0}^{+\infty} p_i f_n\left(x - x_i\right)$$
Q6b Discrete Random Variables Convergence of Expectations or Moments View
Let $K > 0$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ be a positive bounded function with support in $[0, K]$. The function $f$ is the pointwise limit of the sequence $f_n$. Show that the function $f$ satisfies the following equality on $\mathbb{R}$ $$f(x) = g(x) + \sum_{i=0}^{+\infty} p_i f\left(x - x_i\right) \tag{E}$$
Q7a Discrete Random Variables Expectation of a Function of a Discrete Random Variable View
Let $h : \mathbb{R} \rightarrow \mathbb{R}$ be a bounded function that satisfies $h(x) = \sum_{i=0}^{+\infty} p_i h\left(x - x_i\right)$ for all $x \in \mathbb{R}$. Show that for all $x \in \mathbb{R}$ and $n \in \mathbb{N}$, we have $h(x) = \mathbb{E}\left(h\left(x - S_n\right)\right)$.
Q7b Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables View
Let $h : \mathbb{R} \rightarrow \mathbb{R}$ be a bounded function that satisfies $h(x) = \sum_{i=0}^{+\infty} p_i h\left(x - x_i\right)$ for all $x \in \mathbb{R}$, and such that for all $x \in \mathbb{R}$ and $n \in \mathbb{N}$, $h(x) = \mathbb{E}\left(h\left(x - S_n\right)\right)$. Deduce that if moreover the support of $h$ is included in $\mathbb{R}^+$, then for all $x \in \mathbb{R}$, $h(x) = 0$.
Q7c Discrete Random Variables Existence of Expectation or Moments View
Conclude that there exists a unique bounded function with support in $\mathbb{R}^+$ solution of $$f(x) = g(x) + \sum_{i=0}^{+\infty} p_i f\left(x - x_i\right) \tag{E}$$
Q8a Discrete Random Variables Existence of Expectation or Moments View
Show that the set $\Lambda_X := \bigcup_{n \in \mathbb{N}} \left\{y \in \mathbb{R} \mid \mathbb{P}\left(S_n = y\right) > 0\right\}$ is countable and included in $\mathbb{R}^+$.
Q8b Discrete Random Variables Expectation of a Function of a Discrete Random Variable View
We are given an enumeration of the set $\Lambda_X = \{y_i \mid i \in \mathbb{N}\}$. Show that for all $x \in \mathbb{R}$, $$f_n(x) = \sum_{k=0}^{n} \sum_{i=0}^{+\infty} \mathbb{P}\left(S_k = y_i\right) g\left(x - y_i\right)$$
Q8c Discrete Random Variables Expectation and Variance via Combinatorial Counting View
We are given an enumeration of the set $\Lambda_X = \{y_i \mid i \in \mathbb{N}\}$. Deduce that there exists a sequence of positive reals $\left(q_i\right)_{i \geqslant 0}$ such that for all $x \in \mathbb{R}$, $$f(x) = \sum_{i=0}^{+\infty} q_i g\left(x - y_i\right), \quad \text{and} \quad \sum_{i \in \mathbb{N},\, y_i \in [x-K, x]} q_i = \mathbb{E}(N(x-K, x)).$$
Q9a Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
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.
Q9b Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
Suppose that $g$ is continuous. Show that $f$ is uniformly continuous.
Q9c Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
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 $x \in \mathbb{R}$, $$f'(x) = g'(x) + \sum_{i=0}^{+\infty} p_i f'\left(x - x_i\right)$$
Q10a Proof Direct Proof of a Stated Identity or Equality View
Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ such that $$\forall (x, y) \in \Lambda^2, \quad x + y \in \Lambda$$ We say that $\Lambda$ is closed under addition. Show that if $(x, y) \in \Lambda^2$, $(k, n) \in \mathbb{N} \times \mathbb{N}^*$ and $k \leqslant n$, then $nx + k(y-x) \in \Lambda$.
Q10b Proof Existence Proof View
Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition. We define $$\Gamma = \left\{z \in \mathbb{R}_+^* \mid \exists (x, y) \in \Lambda,\, z = y - x\right\}, \quad \text{and} \quad r(\Lambda) = \inf \Gamma.$$ Give two examples of such sets $\Lambda$, one for which $r(\Lambda) > 0$ and another for which $r(\Lambda) = 0$.
Q11a Proof Existence Proof View
Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition, with $$\Gamma = \left\{z \in \mathbb{R}_+^* \mid \exists (x, y) \in \Lambda,\, z = y - x\right\}, \quad r(\Lambda) = \inf \Gamma.$$ We assume that $r(\Lambda) > 0$. Show that there exist $(a, b) \in \Lambda^2$ such that $b - a \in [r(\Lambda), 2r(\Lambda)[$.
Q11b Proof Proof of Set Membership, Containment, or Structural Property View
Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition, with $r(\Lambda) > 0$. Let $(a, b) \in \Lambda^2$ such that $b - a \in [r(\Lambda), 2r(\Lambda)[$ and denote $d = b - a$. Let $k, n \in \mathbb{N}$ such that $k \leqslant n-1$. Show that $$\Lambda \cap [na + kd,\, na + (k+1)d] = \{na + kd,\, na + (k+1)d\}$$
Q11c Proof Existence Proof View
Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition, with $r(\Lambda) > 0$. Let $(a, b) \in \Lambda^2$ such that $b - a \in [r(\Lambda), 2r(\Lambda)[$ and denote $d = b - a$. Show that there exists $n_0 \in \mathbb{N}$ such that $n_0 a + n_0 d > (n_0 + 1)a$, then that there exists $k \in \mathbb{N}$ such that $a = kd$.
Q11d Proof Deduction or Consequence from Prior Results View
Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition, with $r(\Lambda) > 0$ and $d = b - a$ as defined above. Deduce that $\Lambda \subset d\mathbb{Z}$, where $d\mathbb{Z} = \{kd \mid k \in \mathbb{Z}\}$.
Q12a Proof Existence Proof View
Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition, with $r(\Lambda) = 0$. Let $\eta > 0$. Show that there exists $A \geqslant 0$ such that for all $x > A$, $$\Lambda \cap [x, x + \eta] \neq \varnothing$$
Q12b Proof Proof That a Map Has a Specific Property View
Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition, with $r(\Lambda) = 0$. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a uniformly continuous function. Suppose that for every sequence $\left(x_n\right)_{n \geqslant 0}$ with values in $\Lambda$ such that $x_n \rightarrow +\infty$, $f\left(x_n\right) \rightarrow 0$ when $n \rightarrow +\infty$. Show that $f(x) \rightarrow 0$ when $x \rightarrow +\infty$.
Q13a Discrete Probability Distributions Proof of Distributional Properties or Symmetry View
We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$. We consider a function $h$ uniformly continuous and bounded on $\mathbb{R}$ such that for all $x \in \mathbb{R}$, $h(x) \leqslant h(0)$ and $$h(x) = \sum_{i=0}^{+\infty} p_i h\left(x - x_i\right)$$ We recall that for all $x \in \mathbb{R}$ and $n \in \mathbb{N}$, $h(x) = \mathbb{E}\left(h\left(x - S_n\right)\right)$. Show that for all $n \in \mathbb{N}$ and $x \geqslant 0$ such that $\mathbb{P}\left(S_n = x\right) > 0$, we have $h(-x) = h(0)$.
Q13b Number Theory Combinatorial Number Theory and Counting View
We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$. Show that the set $\Lambda_X$ defined in question 8a is closed under addition and that $r\left(\Lambda_X\right) = 0$.
Q13c Proof Deduction or Consequence from Prior Results View
We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$. Using the results of questions 13a and 13b, deduce that $h(-x) \rightarrow h(0)$ when $x \rightarrow +\infty$.
Q13d Proof Deduction or Consequence from Prior Results View
We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$. Conclude that $h$ is a constant function.
Q14a Proof Deduction or Consequence from Prior Results View
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 $K > 0$. The function $f$ is the unique bounded and uniformly continuous solution of equation (E), and $f'$ is bounded and uniformly continuous. Prove that the function $x \mapsto \sup_{t \geqslant x} f'(t)$ admits a finite limit when $x \rightarrow +\infty$. We denote $$c := \lim_{x \rightarrow +\infty} \sup_{t \geqslant x} f'(t)$$
Q14b Proof Existence Proof View
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 $c := \lim_{x \rightarrow +\infty} \sup_{t \geqslant x} f'(t)$, show that there exists a sequence $y_n \rightarrow +\infty$ such that $f'\left(y_n\right) \rightarrow c$ when $n \rightarrow +\infty$.
Q14c Proof Deduction or Consequence from Prior Results View
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 $c$.
Q14d Proof Deduction or Consequence from Prior Results View
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 $c = 0$.
Q14e Proof Deduction or Consequence from Prior Results View
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 $t \rightarrow +\infty$.
Q14f Proof Deduction or Consequence from Prior Results View
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 $t \rightarrow +\infty$.
Q15a Proof Deduction or Consequence from Prior Results View
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 $\ell \geqslant 0$, $$\int_0^{+\infty} g(t)(\mu(t+\ell) - \mu(t))\,dt = 0$$
Q15b Proof Deduction or Consequence from Prior Results View
Under the same assumptions as question 15a, deduce that $\mu$ is constant.
Q16a Proof Direct Proof of a Stated Identity or Equality View
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 $x < 0$.
Q16b Proof Deduction or Consequence from Prior Results View
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 $t \geqslant 0$.
Q17 Sequences and Series Limit Evaluation Involving Sequences View
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. Conclude that for all $g$ of $\mathscr{F}$ piecewise continuous, $$\sum_{k=0}^{+\infty} \mathbb{E}\left(g\left(x - S_k\right)\right) \rightarrow \frac{1}{\mathbb{E}(X)} \int_{-\infty}^{+\infty} g(t)\,dt \quad \text{when} \quad x \rightarrow +\infty$$
Q18 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View
Let $\ell > 0$ be fixed. Determine the behaviour of $\mathbb{E}(N(x, x+\ell))$ when $x \rightarrow +\infty$. Interpret the result. Is this result true if there exists $d > 0$ such that $\mathbb{P}(X \in d\mathbb{Z}) = 1$?