grandes-ecoles

QI.A.1 Sequences and Series Recurrence Relations and Sequence Properties View

Show that the set $E^{c}$ is non-empty.

QI.A.2 Proof Recurrence Relations and Sequence Properties View

Is the set $E^{c}$ a vector subspace of $\mathbb{R}^{\mathbb{N}}$?

QI.A.3 Sequences and Series Recurrence Relations and Sequence Properties View

Show that $E^{c}$ is strictly included in $E$.

QI.A.4 Proof Limit Evaluation Involving Sequences View

Let $\left(u_{n}\right)_{n \in \mathbb{N}}$ be an element of $E^{c}$. Show that $\ell^{c}$ belongs to the segment $[0,1]$.

QI.B.1 Sequences and Series Limit Evaluation Involving Sequences View

Let $k$ be a strictly positive integer and $q$ a real belonging to the interval $]0,1[$. Show that the sequences $\left(\frac{1}{(n+1)^{k}}\right)_{n \in \mathbb{N}},\left(n^{k} q^{n}\right)_{n \in \mathbb{N}}$ and $\left(\frac{1}{n !}\right)_{n \in \mathbb{N}}$ belong to $E^{c}$ and give their convergence rate.

QI.B.2 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View

We consider the sequence $\left(v_{n}\right)_{n \in \mathbb{N}}$ defined by $\forall n \in \mathbb{N}, v_{n}=\left(1+\frac{1}{2^{n}}\right)^{2^{n}}$.
a) Show that in the neighbourhood of $+\infty, v_{n}=\mathrm{e}-\frac{\mathrm{e}}{2^{n+1}}+o\left(\frac{1}{2^{n}}\right)$.
b) Show that the sequence $(v_{n})$ belongs to $E^{c}$ and give its convergence rate.

QI.B.3 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View

We consider the sequence $\left(I_{n}\right)_{n \in \mathbb{N}}$ defined by $I_{0}=0$ and $\forall n \in \mathbb{N}^{*}, I_{n}=\int_{0}^{+\infty} \ln\left(1+\frac{x}{n}\right) \mathrm{e}^{-x} \mathrm{~d} x$.
a) Show that the sequence $(I_{n})$ is well defined and belongs to $E$.
b) Using integration by parts, show that the sequence $(I_{n})$ belongs to $E^{c}$ and give its convergence rate.

QI.B.4 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Let $\alpha$ be a real strictly greater than 1. The Riemann series $\sum_{n \geqslant 1} \frac{1}{n^{\alpha}}$ converges to a real that we will denote $\ell$. We denote by $\left(S_{n}\right)_{n \in \mathbb{N}}$ the sequence defined by $S_{0}=0$ and $\forall n \geqslant 1, S_{n}=\sum_{k=1}^{n} \frac{1}{k^{\alpha}}$.
a) Show that $\forall n \geqslant 1, \frac{1}{\alpha-1} \frac{1}{(n+1)^{\alpha-1}} \leqslant \ell-S_{n} \leqslant \frac{1}{\alpha-1} \frac{1}{n^{\alpha-1}}$.
b) Deduce that $\left(S_{n}\right)_{n \in \mathbb{N}}$ belongs to $E^{c}$ and give its convergence rate.

QI.C.1 Sequences and series, recurrence and convergence Convergence/Divergence Determination of Numerical Series View

Let $\left(u_{n}\right)_{n \in \mathbb{N}}$ be an element of $E$ whose convergence rate is of order $r$, where $r$ is a real strictly greater than 1. Show that the convergence of the sequence $\left(u_{n}\right)_{n \in \mathbb{N}}$ is fast.

QI.C.2 Sequences and Series Convergence/Divergence Determination of Numerical Series View

a) Show that the sequence $\left(S_{n}\right)_{n \in \mathbb{N}}$ defined by $\forall n \in \mathbb{N}, S_{n}=\sum_{k=0}^{n} \frac{1}{k!}$ is an element of $E$. We denote by $s$ the limit of this sequence.
b) Show that for every natural integer $n$, we have $\frac{1}{(n+1)!} \leqslant s-S_{n} \leqslant \frac{1}{(n+1)!} \sum_{k=0}^{+\infty} \frac{1}{2^{k}}$.
c) Deduce that the convergence of the sequence $\left(S_{n}\right)_{n \in \mathbb{N}}$ is fast.
d) Let $r$ be a real strictly greater than 1. Show that the convergence of the sequence $\left(S_{n}\right)_{n \in \mathbb{N}}$ towards $s$ is not of order $r$.

QI.C.3 Fixed Point Iteration View

We consider $I$ a real interval of strictly positive length, $f$ a function defined on $I$ with values in $I$ and $\left(u_{n}\right)_{n \in \mathbb{N}}$ a sequence defined by $u_{0} \in I$ and $\forall n \in \mathbb{N}, u_{n+1}=f\left(u_{n}\right)$. We assume that the sequence $\left(u_{n}\right)_{n \in \mathbb{N}}$ converges to an element $\ell$ of $I$ and that $f$ is differentiable at $\ell$.
a) Show that $f(\ell)=\ell$.
b) Show that if the sequence $\left(u_{n}\right)_{n \in \mathbb{N}}$ is not stationary then it belongs to $E^{c}$. Give its convergence rate as a function of $f^{\prime}(\ell)$.
c) Show that if $\left|f^{\prime}(\ell)\right|>1$, then $\left(u_{n}\right)_{n \in \mathbb{N}}$ is stationary.
d) Let $r$ be an integer greater than or equal to 2. We assume that the function $f$ is of class $\mathcal{C}^{r}$ on $I$ and that the sequence $\left(u_{n}\right)_{n \in \mathbb{N}}$ is not stationary. Show that the convergence rate of $\left(u_{n}\right)_{n \in \mathbb{N}}$ is of order $r$ if and only if $\forall k \in\{1,2, \ldots, r-1\}, f^{(k)}(\ell)=0$.

QII.A.1 Taylor series Construct series for a composite or related function View

We recall that the hyperbolic cosine function, which we denote cosh, is defined, for every real $t$, by $$\cosh(t)=\frac{\mathrm{e}^{t}+\mathrm{e}^{-t}}{2}$$
a) Give the power series expansion of the hyperbolic cosine function and that of the function defined on $\mathbb{R}$ by $t \mapsto \mathrm{e}^{t^{2}/2}$. We will give the radius of convergence of these two power series.
b) Deduce that $\forall t \in \mathbb{R}, \cosh(t) \leqslant \mathrm{e}^{t^{2}/2}$.

QII.A.2 Exponential Functions True/False or Multiple-Statement Verification View

Let $a$ and $b$ be two reals satisfying $a < b$. Show that $\forall \lambda \in [0,1], \mathrm{e}^{\lambda a+(1-\lambda) b} \leqslant \lambda \mathrm{e}^{a}+(1-\lambda) \mathrm{e}^{b}$.

QII.A.3 Curve Sketching Monotonicity and boundedness analysis View

Let $f$ be a function with real values, defined and continuous on $\mathbb{R}^{+}$, and admitting a finite limit at $+\infty$.
a) Show that $f$ is bounded on $\mathbb{R}^{+}$.
b) Deduce that the function $g$ defined on $\mathbb{R}^{+}$ by $\forall t \in \mathbb{R}^{+}, g(t)=t e^{\gamma t}$ where $\gamma$ is a strictly negative real, is bounded on $\mathbb{R}^{+}$.

QII.B.1 Moment generating functions Existence and domain of the MGF View

Let $\alpha$ be a strictly positive real and $X$ a discrete random variable admitting an exponential moment of order $\alpha$. Show that the random variable $e^{\alpha X}$ has finite expectation.

QII.B.2 Moment generating functions Compute MGF or characteristic function for a named distribution View

For each of the following real random variables, determine the strictly positive reals $\alpha$ such that the random variable admits an exponential moment of order $\alpha$ and calculate $\mathbb{E}\left(\mathrm{e}^{\alpha X}\right)$ in this case.
a) $X$ a random variable following a Poisson distribution with parameter $\lambda$, where $\lambda$ is a strictly positive real.
b) $Y$ a random variable following a geometric distribution with parameter $p$, where $p$ is a real strictly between 0 and 1.
c) $Z$ a random variable following a binomial distribution with parameters $n$ and $p$, where $n$ is a strictly positive integer and $p$ is a real strictly between 0 and 1.

QII.C.1 Probability Generating Functions Bounding probabilities or tail estimates via PGF View

In subsection II.C, we consider $\varepsilon$ a strictly positive real, $X$ a discrete real random variable taking values in $\left\{x_{p}, p \in \mathbb{N}\right\}$, and $\left(X_{k}\right)_{k \in \mathbb{N}^{*}}$ a sequence of random variables that are mutually independent and have the same distribution as $X$. For every strictly positive integer $n$, we define the random variable $S_{n}$ by $S_{n}=\sum_{k=1}^{n} X_{k}$. We assume that the random variable $X$ admits an exponential moment of order $\alpha$ where $\alpha$ is a strictly positive real.
a) Show that the variable $X$ has finite expectation. We will denote by $m$ the expectation of $X$.
b) Apply, with appropriate justifications, the weak law of large numbers to the sequence of random variables $\left(X_{k}\right)$.

QII.C.2 Probability Generating Functions Existence and domain of the MGF View

In subsection II.C, we consider $\varepsilon$ a strictly positive real, $X$ a discrete real random variable taking values in $\left\{x_{p}, p \in \mathbb{N}\right\}$, and $\left(X_{k}\right)_{k \in \mathbb{N}^{*}}$ a sequence of random variables that are mutually independent and have the same distribution as $X$. For every strictly positive integer $n$, we define the random variable $S_{n}$ by $S_{n}=\sum_{k=1}^{n} X_{k}$. We assume that the random variable $X$ admits an exponential moment of order $\alpha$ where $\alpha$ is a strictly positive real.
a) Show that the function $\Psi: t \mapsto \mathbb{E}\left(\mathrm{e}^{t X}\right)$ is defined and continuous on the segment $[-\alpha, \alpha]$.
b) Show that the function $\Psi$ is differentiable on the interval $]-\alpha, \alpha[$ and determine its derivative function.

QII.C.3 Probability Generating Functions Extract moments from the MGF or characteristic function View

In subsection II.C, we consider $\varepsilon$ a strictly positive real, $X$ a discrete real random variable taking values in $\left\{x_{p}, p \in \mathbb{N}\right\}$, and $\left(X_{k}\right)_{k \in \mathbb{N}^{*}}$ a sequence of random variables that are mutually independent and have the same distribution as $X$. For every strictly positive integer $n$, we define the random variable $S_{n}$ by $S_{n}=\sum_{k=1}^{n} X_{k}$. We assume that the random variable $X$ admits an exponential moment of order $\alpha$ where $\alpha$ is a strictly positive real. The function $\Psi: t \mapsto \mathbb{E}\left(\mathrm{e}^{tX}\right)$ is defined on $[-\alpha, \alpha]$.
We consider the function $f_{\varepsilon}$ defined by $$f_{\varepsilon}:\left\{\begin{array}{l}[-\alpha, \alpha] \rightarrow \mathbb{R}^{+} \\ t \mapsto \mathrm{e}^{-(m+\varepsilon) t} \Psi(t)\end{array}\right.$$
a) Give the values of $f_{\varepsilon}(0)$ and $f_{\varepsilon}^{\prime}(0)$.
b) Deduce that there exists a real $t_{0}$ belonging to the interval $]0, \alpha[$ satisfying $0 < f_{\varepsilon}\left(t_{0}\right) < 1$.

QII.C.4 Moment generating functions MGF of sums of independent random variables (product property) View

In subsection II.C, we consider $\varepsilon$ a strictly positive real, $X$ a discrete real random variable taking values in $\left\{x_{p}, p \in \mathbb{N}\right\}$, and $\left(X_{k}\right)_{k \in \mathbb{N}^{*}}$ a sequence of random variables that are mutually independent and have the same distribution as $X$. For every strictly positive integer $n$, we define the random variable $S_{n}$ by $S_{n}=\sum_{k=1}^{n} X_{k}$. We assume that the random variable $X$ admits an exponential moment of order $\alpha$ where $\alpha$ is a strictly positive real. The function $\Psi: t \mapsto \mathbb{E}\left(\mathrm{e}^{tX}\right)$ is defined on $[-\alpha, \alpha]$.
Show that for every real $t$ belonging to the segment $[-\alpha, \alpha]$ and every $n$ belonging to $\mathbb{N}^{*}$, the real random variable $\mathrm{e}^{t S_{n}}$ has expectation equal to $(\Psi(t))^{n}$.

QII.C.5 Moment generating functions Concentration inequality via MGF and Markov's inequality (Chernoff method) View

In subsection II.C, we consider $\varepsilon$ a strictly positive real, $X$ a discrete real random variable taking values in $\left\{x_{p}, p \in \mathbb{N}\right\}$, and $\left(X_{k}\right)_{k \in \mathbb{N}^{*}}$ a sequence of random variables that are mutually independent and have the same distribution as $X$. For every strictly positive integer $n$, we define the random variable $S_{n}$ by $S_{n}=\sum_{k=1}^{n} X_{k}$. We assume that the random variable $X$ admits an exponential moment of order $\alpha$ where $\alpha$ is a strictly positive real. The function $\Psi: t \mapsto \mathbb{E}\left(\mathrm{e}^{tX}\right)$ is defined on $[-\alpha, \alpha]$, and $f_{\varepsilon}(t) = \mathrm{e}^{-(m+\varepsilon)t}\Psi(t)$.
a) Let $t$ be a real belonging to the interval $]0, \alpha]$ and let $n$ belong to $\mathbb{N}^{*}$. Show that $\mathbb{P}\left(\frac{S_{n}}{n} \geqslant m+\varepsilon\right)=\mathbb{P}\left(\mathrm{e}^{t S_{n}} \geqslant\left(\mathrm{e}^{t(m+\varepsilon)}\right)^{n}\right)$, then that $\mathbb{P}\left(\frac{S_{n}}{n} \geqslant m+\varepsilon\right) \leqslant\left(f_{\varepsilon}(t)\right)^{n}$.
b) Deduce that there exists a real $r$ belonging to the interval $]0,1[$ such that $\forall n \in \mathbb{N}^{*}, \mathbb{P}\left(\frac{S_{n}}{n} \geqslant m+\varepsilon\right) \leqslant r^{n}$.

QII.C.6 Moment generating functions Concentration inequality via MGF and Markov's inequality (Chernoff method) View

In subsection II.C, we consider $\varepsilon$ a strictly positive real, $X$ a discrete real random variable taking values in $\left\{x_{p}, p \in \mathbb{N}\right\}$, and $\left(X_{k}\right)_{k \in \mathbb{N}^{*}}$ a sequence of random variables that are mutually independent and have the same distribution as $X$. For every strictly positive integer $n$, we define the random variable $S_{n}$ by $S_{n}=\sum_{k=1}^{n} X_{k}$. We assume that the random variable $X$ admits an exponential moment of order $\alpha$ where $\alpha$ is a strictly positive real, and $m = \mathbb{E}(X)$.
Show that the sequence defined by: $\forall n \in \mathbb{N}^{*}, \mathbb{P}\left(\left|\frac{S_{n}}{n}-m\right| \geqslant \varepsilon\right)$ is bounded above by a sequence with limit zero and whose convergence rate is geometric. Compare this result to the upper bound obtained with the weak law of large numbers.

QII.D.1 Moment generating functions Existence and domain of the MGF View

In subsection II.D, we assume that there exists a strictly positive real number $c$ such that the discrete real random variable $X$ satisfies $\mathbb{E}(X)=0$ and $\forall \omega \in \Omega,|X(\omega)| \leqslant c$.
Show that the random variable $X$ admits an exponential moment of order $\alpha$ for every strictly positive real number $\alpha$.

QII.D.2 Moment generating functions Upper bound on MGF (sub-Gaussian or exponential inequalities) View

In subsection II.D, we assume that there exists a strictly positive real number $c$ such that the discrete real random variable $X$ satisfies $\mathbb{E}(X)=0$ and $\forall \omega \in \Omega,|X(\omega)| \leqslant c$.
We consider $Y$ the real random variable defined by $Y=\frac{1}{2}-\frac{X}{2c}$.
a) Verify that $X=-cY+(1-Y)c$.
b) Show that $\mathrm{e}^{X} \leqslant Y \mathrm{e}^{-c}+(1-Y) \mathrm{e}^{c}$.

QII.D.3 Moment generating functions Upper bound on MGF (sub-Gaussian or exponential inequalities) View

In subsection II.D, we assume that there exists a strictly positive real number $c$ such that the discrete real random variable $X$ satisfies $\mathbb{E}(X)=0$ and $\forall \omega \in \Omega,|X(\omega)| \leqslant c$.
a) Show that $\mathbb{E}\left(\mathrm{e}^{X}\right) \leqslant \cosh(c)$.
b) Deduce that $\forall t \in \mathbb{R}^{+*}, \Psi(t) \leqslant \cosh(ct)$.

QII.D.4 Moment generating functions Upper bound on MGF (sub-Gaussian or exponential inequalities) View

In subsection II.D, we assume that there exists a strictly positive real number $c$ such that the discrete real random variable $X$ satisfies $\mathbb{E}(X)=0$ and $\forall \omega \in \Omega,|X(\omega)| \leqslant c$. The functions $\Psi$ and $f_{\varepsilon}$ are defined on $\mathbb{R}$, with $f_{\varepsilon}(t) = \mathrm{e}^{-\varepsilon t}\Psi(t)$ (since $m=0$).
Show that $\forall t \in \mathbb{R}^{+*}, f_{\varepsilon}(t) \leqslant \exp\left(-t\varepsilon+\frac{1}{2}c^{2}t^{2}\right)$.

QII.D.5 Central limit theorem Concentration inequality via MGF and Markov's inequality (Chernoff method) View

In subsection II.D, we assume that there exists a strictly positive real number $c$ such that the discrete real random variable $X$ satisfies $\mathbb{E}(X)=0$ and $\forall \omega \in \Omega,|X(\omega)| \leqslant c$. For every strictly positive integer $n$, $S_{n}=\sum_{k=1}^{n} X_{k}$ where $\left(X_{k}\right)$ are mutually independent with the same distribution as $X$.
Show that $\forall n \in \mathbb{N}^{*}, \mathbb{P}\left(\left|\frac{S_{n}}{n}\right| \geqslant \varepsilon\right) \leqslant 2 \exp\left(-n \frac{\varepsilon^{2}}{2c^{2}}\right)$.

QII.D.6 Central limit theorem Derive or Prove a Binomial Distribution Identity View

In subsection II.D, we assume that there exists a strictly positive real number $c$ such that the discrete real random variable $X$ satisfies $\mathbb{E}(X)=0$ and $\forall \omega \in \Omega,|X(\omega)| \leqslant c$. We have shown that $\forall n \in \mathbb{N}^{*}, \mathbb{P}\left(\left|\frac{S_{n}}{n}\right| \geqslant \varepsilon\right) \leqslant 2 \exp\left(-n \frac{\varepsilon^{2}}{2c^{2}}\right)$.
Let $n$ be a non-zero natural number, $p$ an element of the interval $]0,1[$ and $Z$ a random variable following a binomial distribution with parameter $(n, p)$. Using the previous question, bound $\mathbb{P}\left(\left|\frac{Z}{n}-p\right| \geqslant \varepsilon\right)$ as a function of $n, p$ and $\varepsilon$.

grandes-ecoles

Papers (176)

2017 centrale-maths2__pc