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Papers (191)
2025
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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 centrale-maths2__psi

27 maths questions

QI.B Taylor series Prove smoothness or power series expandability of a function View
We consider the function $\psi$ defined on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}^{*}, \quad \psi(x) = \frac{\sin(\pi x)}{\pi x} \quad \text{and} \quad \psi(0) = 1$$
I.B.1) Justify that $\psi$ is expandable as a power series. Specify this expansion and its radius of convergence. Deduce that $\psi$ is of class $C^{\infty}$ on $\mathbb{R}$.
I.B.2) Prove that
$$\forall n \in \mathbb{N}, \quad \int_{n}^{n+1} |\psi(x)| \mathrm{d}x \geqslant \frac{2}{(n+1)\pi^{2}}$$
Deduce that $\psi$ does not belong to $E_{\mathrm{cpm}}$.
QI.C Indefinite & Definite Integrals Properties of Integral-Defined Functions (Continuity, Differentiability) View
Let $f \in E_{\mathrm{cpm}}$. Show that the function $\mathcal{F}(f)$ is continuous on $\mathbb{R}$.
QI.D Taylor series Prove smoothness or power series expandability of a function View
Let $f \in \mathcal{S}$.
I.D.1) Justify that, for every natural number $n$, the function $x \mapsto x^{n} f(x)$ is integrable on $\mathbb{R}$.
I.D.2) Prove that the function $\mathcal{F}(f)$ is of class $C^{\infty}$ on $\mathbb{R}$ and that
$$\forall n \in \mathbb{N}, \quad \forall \xi \in \mathbb{R}, \quad (\mathcal{F}(f))^{(n)}(\xi) = (-2\pi\mathrm{i})^{n} \int_{-\infty}^{+\infty} t^{n} f(t) e^{-2\pi\mathrm{i} t\xi} \mathrm{~d}t$$
QI.E First order differential equations (integrating factor) View
We consider the function $\theta : \mathbb{R} \rightarrow \mathbb{C}$ defined by $\theta(x) = \exp(-\pi x^{2})$, for $x \in \mathbb{R}$.
I.E.1) Justify that $\theta$ belongs to $\mathcal{S}$ and that $\mathcal{F}(\theta)$ is a solution of the differential equation
$$\forall \xi \in \mathbb{R}, \quad y'(\xi) = -2\pi\xi\, y(\xi)$$
I.E.2) Establish that $\mathcal{F}(\theta) = \theta$.
We will admit that $\int_{-\infty}^{+\infty} \theta(x) \mathrm{d}x = 1$.
QII.A Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
Let $f \in \mathcal{S}$. We assume that $\mathcal{F}(f)$ is integrable on $\mathbb{R}$. For every positive natural number $n$, we set
$$I_{n} = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) \theta\left(\frac{\xi}{n}\right) \mathrm{d}\xi \quad J_{n} = \int_{-\infty}^{+\infty} f\left(\frac{t}{n}\right) \mathcal{F}(\theta)(t) \mathrm{d}t$$
Show that $\lim_{n \rightarrow +\infty} I_{n} = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) \mathrm{d}\xi$.
QII.B Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
Let $f \in \mathcal{S}$. We assume that $\mathcal{F}(f)$ is integrable on $\mathbb{R}$. For every positive natural number $n$, we set
$$I_{n} = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) \theta\left(\frac{\xi}{n}\right) \mathrm{d}\xi \quad J_{n} = \int_{-\infty}^{+\infty} f\left(\frac{t}{n}\right) \mathcal{F}(\theta)(t) \mathrm{d}t$$
Calculate $\lim_{n \rightarrow +\infty} J_{n}$.
QII.C Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
Let $f \in \mathcal{S}$. We assume that $\mathcal{F}(f)$ is integrable on $\mathbb{R}$. For every positive natural number $n$, we set
$$I_{n} = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) \theta\left(\frac{\xi}{n}\right) \mathrm{d}\xi \quad J_{n} = \int_{-\infty}^{+\infty} f\left(\frac{t}{n}\right) \mathcal{F}(\theta)(t) \mathrm{d}t$$
Prove that $\forall n \in \mathbb{N}^{*}, I_{n} = J_{n}$.
We will admit the Fubini formula:
$$\int_{-\infty}^{+\infty} \left(\int_{-\infty}^{+\infty} f(t) \theta\left(\frac{\xi}{n}\right) e^{-2\pi\mathrm{i} t\xi} \mathrm{d}\xi\right) \mathrm{d}t = \int_{-\infty}^{+\infty} \left(\int_{-\infty}^{+\infty} f(t) \theta\left(\frac{\xi}{n}\right) e^{-2\pi\mathrm{i} t\xi} \mathrm{d}t\right) \mathrm{d}\xi$$
QII.D Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View
Let $f \in \mathcal{S}$. We assume that $\mathcal{F}(f)$ is integrable on $\mathbb{R}$.
Prove that $f(0) = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) \mathrm{d}\xi$.
Deduce, using the function $h : t \mapsto f(x+t)$, that
$$\forall x \in \mathbb{R}, \quad f(x) = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x\xi} \mathrm{d}\xi$$
QII.E First order differential equations (integrating factor) View
Prove that $\forall x \in \mathbb{R}, \quad \int_{-\infty}^{+\infty} \frac{e^{2\pi\mathrm{i} x\xi}}{1+(2\pi\xi)^{2}} \mathrm{d}\xi = \frac{1}{2} e^{-|x|}$.
QIII.A Taylor series Prove smoothness or power series expandability of a function View
Let $f$ be a function in $\mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. According to the Fourier inversion formula, we have
$$\forall x \in \mathbb{R}, \quad f(x) = \int_{-1/2}^{1/2} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x\xi} \mathrm{d}\xi$$
Prove that $\mathcal{F}(f)$ is of class $C^{\infty}$ on $\mathbb{R}$ and that $\mathcal{F}(f) \in \mathcal{S}$. Deduce that $f$ is of class $C^{\infty}$ on $\mathbb{R}$.
QIII.B Taylor series Prove smoothness or power series expandability of a function View
Let $f$ be a function in $\mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. According to the Fourier inversion formula, we have
$$\forall x \in \mathbb{R}, \quad f(x) = \int_{-1/2}^{1/2} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x\xi} \mathrm{d}\xi$$
Prove that
$$\forall (x, x_{0}) \in \mathbb{R}^{2}, \quad \sum_{n=0}^{+\infty} \frac{(x-x_{0})^{n}}{n!} \int_{-1/2}^{1/2} (2\pi\mathrm{i}\xi)^{n} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x_{0}\xi} \mathrm{d}\xi = f(x)$$
QIII.C Taylor series Prove smoothness or power series expandability of a function View
Let $f$ be a function in $\mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. According to the Fourier inversion formula, we have
$$\forall x \in \mathbb{R}, \quad f(x) = \int_{-1/2}^{1/2} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x\xi} \mathrm{d}\xi$$
We assume that $f$ is zero outside a segment $[a, b]$. Show that $f = 0$.
QIV.A Differentiating Transcendental Functions Regularity and smoothness of transcendental functions View
For every natural number $n$, we denote by $S_{n}$ the function defined on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}, \quad S_{n}(x) = \sum_{k=-n}^{n} e^{2\pi\mathrm{i} kx}$$
Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. We consider the function $g$ defined on $[-1,1]$ by
$$\forall x \in ]-1,1[\backslash\{0\}, \quad g(x) = \frac{f(x)-f(0)}{\sin(\pi x)} \quad g(0) = \frac{f'(0)}{\pi} \quad g(1) = g(-1) = -g(0)$$
and the sequence of complex numbers $(c_{n}(f))_{n \in \mathbb{Z}}$ defined by
$$\forall n \in \mathbb{Z}, \quad c_{n}(f) = \int_{-1/2}^{1/2} f(x) e^{-2\pi\mathrm{i} nx} \mathrm{d}x$$
IV.A.1) Show that the function $g$ is of class $C^{1}$ on $]-1,1[\backslash\{0\}$ and continuous on $]-1,1[$.
IV.A.2) Calculate the limit of $g'$ at 0. Deduce that $g$ is of class $C^{1}$ on $]-1,1[$.
QIV.B Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
For every natural number $n$, we denote by $S_{n}$ the function defined on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}, \quad S_{n}(x) = \sum_{k=-n}^{n} e^{2\pi\mathrm{i} kx}$$
Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. The sequence of complex numbers $(c_{n}(f))_{n \in \mathbb{Z}}$ is defined by
$$\forall n \in \mathbb{Z}, \quad c_{n}(f) = \int_{-1/2}^{1/2} f(x) e^{-2\pi\mathrm{i} nx} \mathrm{d}x$$
Let $n \in \mathbb{N}$. Calculate the integral $\int_{-1/2}^{1/2} S_{n}(x) \mathrm{d}x$.
QIV.C Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
For every natural number $n$, we denote by $S_{n}$ the function defined on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}, \quad S_{n}(x) = \sum_{k=-n}^{n} e^{2\pi\mathrm{i} kx}$$
Prove that
$$\forall n \in \mathbb{N}, \quad \forall x \in \left[-\frac{1}{2}, \frac{1}{2}\right] \backslash\{0\}, \quad S_{n}(x) = \frac{\sin((2n+1)\pi x)}{\sin(\pi x)}$$
QIV.D Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
For every natural number $n$, we denote by $S_{n}$ the function defined on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}, \quad S_{n}(x) = \sum_{k=-n}^{n} e^{2\pi\mathrm{i} kx}$$
Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. We consider the function $g$ defined on $[-1,1]$ by
$$\forall x \in ]-1,1[\backslash\{0\}, \quad g(x) = \frac{f(x)-f(0)}{\sin(\pi x)} \quad g(0) = \frac{f'(0)}{\pi} \quad g(1) = g(-1) = -g(0)$$
and the sequence of complex numbers $(c_{n}(f))_{n \in \mathbb{Z}}$ defined by
$$\forall n \in \mathbb{Z}, \quad c_{n}(f) = \int_{-1/2}^{1/2} f(x) e^{-2\pi\mathrm{i} nx} \mathrm{d}x$$
Justify that
$$\forall n \in \mathbb{N}^{*}, \quad \sum_{k=-n}^{n} c_{k}(f) = f(0) + \int_{-1/2}^{1/2} g(x) \sin((2n+1)\pi x) \mathrm{d}x$$
QIV.E Integration by Parts Prove an Integral Inequality or Bound View
Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. We consider the function $g$ defined on $[-1,1]$ by
$$\forall x \in ]-1,1[\backslash\{0\}, \quad g(x) = \frac{f(x)-f(0)}{\sin(\pi x)} \quad g(0) = \frac{f'(0)}{\pi} \quad g(1) = g(-1) = -g(0)$$
We henceforth admit that $g$ is of class $C^{1}$ on $[-1,1]$.
Using integration by parts, show the existence of a real number $C$ such that
$$\forall n \in \mathbb{N}, \quad \left|\int_{-1/2}^{1/2} g(x) \sin((2n+1)\pi x) \mathrm{d}x\right| \leqslant \frac{C}{2n+1}$$
QIV.F Differentiating Transcendental Functions Prove inequality or sign of transcendental expression View
Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. Let $t \in [-1/2, 1/2]$. We consider the function $G_{t}$ defined on $[-1/2, 1/2]$ by
$$\forall x \in \left[-\frac{1}{2}, \frac{1}{2}\right], \quad G_{t}(x) = f'(x+t)\sin(\pi x) - (f(x+t)-f(t))\pi\cos(\pi x)$$
Establish the existence of a real number $D$, independent of $x$ and $t$, such that
$$\forall t \in \left[-\frac{1}{2}, \frac{1}{2}\right], \quad \forall x \in \left[-\frac{1}{2}, \frac{1}{2}\right], \quad |G_{t}(x)| \leqslant D x^{2}$$
QIV.G Taylor series Lagrange error bound application View
Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. The sequence of complex numbers $(c_{n}(f))_{n \in \mathbb{Z}}$ is defined by
$$\forall n \in \mathbb{Z}, \quad c_{n}(f) = \int_{-1/2}^{1/2} f(x) e^{-2\pi\mathrm{i} nx} \mathrm{d}x$$
Prove the existence of a real number $E$ such that
$$\forall t \in \left[-\frac{1}{2}, \frac{1}{2}\right], \quad \left|f(t) - \sum_{k=-n}^{n} c_{k}(f) e^{2\pi\mathrm{i} kt}\right| \leqslant \frac{E}{2n+1}$$
One may introduce the function $h_{t} : x \mapsto f(x+t)$.
QV.A Taylor series Prove smoothness or power series expandability of a function View
Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. We set
$$\forall k \in \mathbb{Z}, \quad \forall x \in \mathbb{R}, \quad \psi_{k}(x) = \psi(x+k)$$
where $\psi(x) = \frac{\sin(\pi x)}{\pi x}$ for $x \neq 0$ and $\psi(0) = 1$.
Justify that $\forall n \in \mathbb{N}, \quad (\mathcal{F}(f))^{(n)}\left(\frac{1}{2}\right) = (\mathcal{F}(f))^{(n)}\left(-\frac{1}{2}\right) = 0$.
QV.B Differentiating Transcendental Functions Regularity and smoothness of transcendental functions View
Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. Let $h$ be the function defined on $\mathbb{R}$, which is 1-periodic and which equals $\mathcal{F}(f)$ on the interval $[-1/2, 1/2]$. Show that $h$ is of class $C^{\infty}$ on $\mathbb{R}$.
QV.C Taylor series Prove smoothness or power series expandability of a function View
Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. Let $h$ be the function defined on $\mathbb{R}$, which is 1-periodic and which equals $\mathcal{F}(f)$ on the interval $[-1/2, 1/2]$.
Using the inequality from IV.G, prove the existence of a sequence of complex numbers $(d_{k})_{k \in \mathbb{Z}}$ such that the sequence of functions $\left(x \mapsto \sum_{k=-n}^{n} d_{k} e^{2\pi\mathrm{i} kx}\right)_{n \in \mathbb{N}}$ converges uniformly to $\mathcal{F}(f)$ on $[-1/2, 1/2]$.
QV.D Taylor series Prove smoothness or power series expandability of a function View
Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. We set
$$\forall k \in \mathbb{Z}, \quad \forall x \in \mathbb{R}, \quad \psi_{k}(x) = \psi(x+k)$$
where $\psi(x) = \frac{\sin(\pi x)}{\pi x}$ for $x \neq 0$ and $\psi(0) = 1$.
Let $(d_{k})_{k \in \mathbb{Z}}$ be the sequence of complex numbers from V.C. Prove that the sequence of functions $\left(\sum_{k=-n}^{n} d_{k} \psi_{k}\right)_{n \in \mathbb{N}}$ converges uniformly to $f$ on $\mathbb{R}$.
QV.E Taylor series Prove smoothness or power series expandability of a function View
Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. We set
$$\forall k \in \mathbb{Z}, \quad \forall x \in \mathbb{R}, \quad \psi_{k}(x) = \psi(x+k)$$
where $\psi(x) = \frac{\sin(\pi x)}{\pi x}$ for $x \neq 0$ and $\psi(0) = 1$, and $f = \sum_{k=-\infty}^{+\infty} d_{k} \psi_{k}$ (uniform limit).
Establish that $\forall j \in \mathbb{Z},\ f(-j) = d_{j}$.
QVI.A Poisson distribution View
We consider $(X_{n})_{n \in \mathbb{N}^{*}}$ a sequence of random variables defined on the same probability space $(\Omega, \mathcal{A}, P)$, mutually independent and following the same Poisson distribution with parameter $\lambda > 0$. We set
$$\forall n \in \mathbb{N}^{*}, \quad S_{n} = X_{1} + \cdots + X_{n}$$
VI.A.1) By induction, prove that, for every integer $n \in \mathbb{N}$, $S_{n}$ follows the Poisson distribution with parameter $n\lambda$.
We will admit that, for every integer $n \in \mathbb{N}^{*}$, the variables $S_{n}$ and $X_{n+1}$ are mutually independent.
VI.A.2) Let $\varepsilon \in \mathbb{R}_{+}^{*}$. Prove that
$$\forall n \in \mathbb{N}^{*}, \quad P\left(\left|S_{n} - n\lambda\right| \geqslant n\varepsilon\right) \leqslant \frac{\lambda}{n\varepsilon^{2}}$$
VI.A.3) Let $\varepsilon > 0$. Justify the following two inclusions
$$\begin{aligned} & \left(S_{n} > n(\lambda+\varepsilon)\right) \subset \left(\left|S_{n}-n\lambda\right| \geqslant n\varepsilon\right) \\ & \left(S_{n} \leqslant n(\lambda-\varepsilon)\right) \subset \left(\left|S_{n}-n\lambda\right| \geqslant n\varepsilon\right) \end{aligned}$$
VI.A.4) In all the following questions, we assume $x \geqslant 0$.
Deduce from VI.A.3 that
$$\begin{cases} \lim_{n \rightarrow +\infty} P\left(S_{n} \leqslant nx\right) = 0 & \text{if } 0 \leqslant x < \lambda \\ \lim_{n \rightarrow +\infty} P\left(S_{n} \leqslant nx\right) = 1 & \text{if } x > \lambda \end{cases}$$
QVI.B Poisson distribution View
We consider $(X_{n})_{n \in \mathbb{N}^{*}}$ a sequence of random variables defined on the same probability space $(\Omega, \mathcal{A}, P)$, mutually independent and following the same Poisson distribution with parameter $\lambda > 0$. We set $S_{n} = X_{1} + \cdots + X_{n}$.
Using question VI.A, show that
$$\lim_{n \rightarrow +\infty} \sum_{0 \leqslant k \leqslant \lfloor nx \rfloor} \frac{(n\lambda)^{k}}{k!} e^{-n\lambda} = \begin{cases} 0 & \text{if } 0 \leqslant x < \lambda \\ 1 & \text{if } x > \lambda \end{cases}$$
QVI.C First order differential equations (integrating factor) View
Let $f : \mathbb{R}_{+} \rightarrow \mathbb{C}$ be a continuous function and zero outside a segment. We define the function $\mathcal{L}(f)$ (Laplace transform of $f$) on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}, \quad \mathcal{L}(f)(x) = \int_{0}^{+\infty} f(t) e^{-xt} \mathrm{d}t$$
We will admit that $\mathcal{L}(f)$ is of class $C^{\infty}$ on $\mathbb{R}$ and that
$$\forall x \in \mathbb{R}, \quad \forall n \in \mathbb{N}, \quad (\mathcal{L}(f))^{(n)}(x) = (-1)^{n} \int_{0}^{+\infty} f(t) t^{n} e^{-xt} \mathrm{d}t$$
In the remainder of this part, we shall assume that
$$\lim_{n \rightarrow +\infty} \sum_{0 \leqslant k \leqslant \lfloor nx \rfloor} \frac{(n\lambda)^{k}}{k!} e^{-n\lambda} = \frac{1}{2} \quad \text{if } x = \lambda$$
VI.C.1) Let $x \in \mathbb{R}_{+}$. Prove that
$$\lim_{n \rightarrow +\infty} \sum_{0 \leqslant k \leqslant \lfloor nx \rfloor} (-1)^{k} \frac{n^{k}}{k!} (\mathcal{L}(f))^{(k)}(n) = \int_{0}^{x} f(y) \mathrm{d}y$$
VI.C.2) Deduce that the map $\mathcal{L} : f \mapsto \mathcal{L}(f)$ is injective on the set of complex-valued functions, continuous on $\mathbb{R}_{+}$ and zero outside a bounded interval.