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
2021 centrale-maths1__mp

40 maths questions

Q1 Matrices Matrix Norm, Convergence, and Inequality View
Show that, for every $M$ in $\mathcal{M}_{n}(\mathbb{R})$ and for all $P$ and $Q$ in $\mathcal{O}_{n}(\mathbb{R})$, we have $\|PMQ\|_{F} = \|M\|_{F}$.
Q2 Matrices Diagonalizability and Similarity View
We denote $D_{A} = \operatorname{diag}\left(\lambda_{1}(A), \ldots, \lambda_{n}(A)\right)$ and $D_{B} = \operatorname{diag}\left(\lambda_{1}(B), \ldots, \lambda_{n}(B)\right)$. Show that there exists an orthogonal matrix $P = \left(p_{i,j}\right)_{1 \leqslant i,j \leqslant n}$ such that $\|A - B\|_{F}^{2} = \left\|D_{A}P - PD_{B}\right\|_{F}^{2}$.
Q3 Matrices Matrix Entry and Coefficient Identities View
Show that $$\|A - B\|_{F}^{2} = \sum_{1 \leqslant i,j \leqslant n} p_{i,j}^{2}\left(\lambda_{i}(A) - \lambda_{j}(B)\right)^{2}.$$
Q4 Matrices Matrix Norm, Convergence, and Inequality View
We denote $\mathcal{B}_{n}(\mathbb{R})$ the set of doubly stochastic matrices in $\mathcal{M}_{n}(\mathbb{R})$, that is the set of matrices $M = \left(m_{i,j}\right)_{1 \leqslant i,j \leqslant n}$ whose coefficients are all non-negative and such that $\sum_{j=1}^{n} m_{i,j} = \sum_{j=1}^{n} m_{j,i} = 1$ for every $i \in \llbracket 1, n \rrbracket$.
We denote $f : \left|\, \begin{array}{ccc} \mathcal{M}_{n}(\mathbb{R}) & \rightarrow & \mathbb{R} \\ M & \mapsto & \sum_{1 \leqslant i,j \leqslant n} m_{i,j}\left(\lambda_{i}(A) - \lambda_{j}(B)\right)^{2}. \end{array}\right.$
Justify that $f$ admits a minimum on $\mathcal{B}_{n}(\mathbb{R})$.
Q5 Matrices Matrix Norm, Convergence, and Inequality View
We denote $\mathcal{B}_{n}(\mathbb{R})$ the set of doubly stochastic matrices in $\mathcal{M}_{n}(\mathbb{R})$, that is the set of matrices $M = \left(m_{i,j}\right)_{1 \leqslant i,j \leqslant n}$ whose coefficients are all non-negative and such that $\sum_{j=1}^{n} m_{i,j} = \sum_{j=1}^{n} m_{j,i} = 1$ for every $i \in \llbracket 1, n \rrbracket$.
We denote $f : \left|\, \begin{array}{ccc} \mathcal{M}_{n}(\mathbb{R}) & \rightarrow & \mathbb{R} \\ M & \mapsto & \sum_{1 \leqslant i,j \leqslant n} m_{i,j}\left(\lambda_{i}(A) - \lambda_{j}(B)\right)^{2}. \end{array}\right.$
Let $(i,j,k) \in \llbracket 1,n \rrbracket^{3}$ such that $j \geqslant i$ and $k \geqslant i$. Show that, for $M \in \mathcal{M}_{n}(\mathbb{R})$ and for $x \in \mathbb{R}^{+}$, $$f\left(M + xE_{ii} + xE_{jk} - xE_{ik} - xE_{ji}\right) - f(M) = 2x\left(\lambda_{i}(A) - \lambda_{j}(A)\right)\left(\lambda_{k}(B) - \lambda_{i}(B)\right) \leqslant 0$$
Q6 Matrices Matrix Norm, Convergence, and Inequality View
We denote $\mathcal{B}_{n}(\mathbb{R})$ the set of doubly stochastic matrices in $\mathcal{M}_{n}(\mathbb{R})$, that is the set of matrices $M = \left(m_{i,j}\right)_{1 \leqslant i,j \leqslant n}$ whose coefficients are all non-negative and such that $\sum_{j=1}^{n} m_{i,j} = \sum_{j=1}^{n} m_{j,i} = 1$ for every $i \in \llbracket 1, n \rrbracket$.
We denote $f : \left|\, \begin{array}{ccc} \mathcal{M}_{n}(\mathbb{R}) & \rightarrow & \mathbb{R} \\ M & \mapsto & \sum_{1 \leqslant i,j \leqslant n} m_{i,j}\left(\lambda_{i}(A) - \lambda_{j}(B)\right)^{2}. \end{array}\right.$
Let $n \geqslant 2$ and $M = \left(m_{i,j}\right)_{1 \leqslant i,j \leqslant n} \in \mathcal{B}_{n}(\mathbb{R})$ a matrix different from the identity. We denote $i$ the smallest integer belonging to $\llbracket 1,n \rrbracket$ such that $m_{i,i} \neq 1$. Show that there exists a matrix $M^{\prime} = \left(m_{i,j}^{\prime}\right)_{1 \leqslant i,j \leqslant n} \in \mathcal{B}_{n}(\mathbb{R})$ such that $f\left(M^{\prime}\right) \leqslant f(M)$ and $m_{j,j}^{\prime} = 1$ for every $j \in \llbracket 1,i \rrbracket$.
Q7 Matrices Matrix Norm, Convergence, and Inequality View
We denote $\mathcal{B}_{n}(\mathbb{R})$ the set of doubly stochastic matrices in $\mathcal{M}_{n}(\mathbb{R})$, that is the set of matrices $M = \left(m_{i,j}\right)_{1 \leqslant i,j \leqslant n}$ whose coefficients are all non-negative and such that $\sum_{j=1}^{n} m_{i,j} = \sum_{j=1}^{n} m_{j,i} = 1$ for every $i \in \llbracket 1, n \rrbracket$.
We denote $f : \left|\, \begin{array}{ccc} \mathcal{M}_{n}(\mathbb{R}) & \rightarrow & \mathbb{R} \\ M & \mapsto & \sum_{1 \leqslant i,j \leqslant n} m_{i,j}\left(\lambda_{i}(A) - \lambda_{j}(B)\right)^{2}. \end{array}\right.$
Deduce that $$\min\left\{f(M) \mid M \in \mathcal{B}_{n}(\mathbb{R})\right\} = f\left(I_{n}\right)$$
Q8 Matrices Matrix Norm, Convergence, and Inequality View
Deduce that $$\forall (A,B) \in \mathcal{S}_{n}(\mathbb{R})^{2}, \quad \sum_{i=1}^{n}\left(\lambda_{i}(A) - \lambda_{i}(B)\right)^{2} \leqslant \|A - B\|_{F}^{2}.$$
Q9 Permutations & Arrangements Combinatorial Proof or Identity Derivation View
For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convention $C_{0} = 1$.
By enumerating the different well-parenthesized words of length 2, 4 and 6, show that $C_{1} = 1$, $C_{2} = 2$ and determine $C_{3}$.
Q10 Sequences and Series Power Series Expansion and Radius of Convergence View
For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convention $C_{0} = 1$.
Show that, for every natural integer $n$, $C_{n} \leqslant 2^{2n}$. What can we deduce for the radius of convergence of the power series $\sum C_{k} x^{k}$?
Q11 Permutations & Arrangements Combinatorial Proof or Identity Derivation View
For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convention $C_{0} = 1$.
Show by a combinatorial argument that, for every integer $k \geqslant 1$, $$C_{k} = \sum_{i=0}^{k-1} C_{i} C_{k-i-1}$$ We can note that a well-parenthesized word is necessarily of the form $(m)m^{\prime}$ with $m$ and $m^{\prime}$ two well-parenthesized words, possibly empty.
Q12 Sequences and Series Functional Equations and Identities via Series View
For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convention $C_{0} = 1$. For every $x \in ]-\frac{1}{4}, \frac{1}{4}[$, we set $F(x) = \sum_{k=0}^{+\infty} C_{k} x^{k}$.
Show that, for every $x \in ]-\frac{1}{4}, \frac{1}{4}[$, $F(x) = 1 + x(F(x))^{2}$.
Q13 Sequences and series, recurrence and convergence Series convergence and power series analysis View
For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convention $C_{0} = 1$. For every $x \in ]-\frac{1}{4}, \frac{1}{4}[$, we set $F(x) = \sum_{k=0}^{+\infty} C_{k} x^{k}$.
Show that the function $f : \left|]-\frac{1}{4}, \frac{1}{4}[ \begin{array}{cc} & \rightarrow \\ x & \mathbb{R} \\ & \mapsto 2xF(x) - 1 \end{array}\right.$ does not vanish.
Q14 Sequences and series, recurrence and convergence Closed-form expression derivation View
For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convention $C_{0} = 1$. For every $x \in ]-\frac{1}{4}, \frac{1}{4}[$, we set $F(x) = \sum_{k=0}^{+\infty} C_{k} x^{k}$.
Determine, for every $x \in ]-\frac{1}{4}, \frac{1}{4}[$, an expression of $F(x)$ as a function of $x$.
Q15 Generalised Binomial Theorem View
Determine the power series expansion of the function $u \mapsto \sqrt{1-u}$. We will write the coefficients as a quotient of factorials and powers of 2.
Q16 Number Theory Divisibility and Divisor Analysis View
Show that, for every natural integer $n$, $$C_{n} = \frac{(2n)!}{(n+1)! \, n!}$$
Q17 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
For every natural integer $k$ we set $$m_{k} = \frac{1}{2\pi} \int_{-2}^{2} x^{k} \sqrt{4 - x^{2}} \, \mathrm{d}x$$ For $k \in \mathbb{N}$, what is the value of $m_{2k+1}$?
Q18 Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View
For every natural integer $k$ we set $$m_{k} = \frac{1}{2\pi} \int_{-2}^{2} x^{k} \sqrt{4 - x^{2}} \, \mathrm{d}x$$ Using the change of variable $x = 2\sin t$, calculate $m_{0}$.
Q19 Integration by Parts Reduction Formula or Recurrence via Integration by Parts View
For every natural integer $k$ we set $$m_{k} = \frac{1}{2\pi} \int_{-2}^{2} x^{k} \sqrt{4 - x^{2}} \, \mathrm{d}x$$ Using integration by parts, show that, for every natural integer $k$, $$m_{2k+2} = \frac{2(2k+1)}{k+2} m_{2k}$$
Q20 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
For every natural integer $k$ we set $$m_{k} = \frac{1}{2\pi} \int_{-2}^{2} x^{k} \sqrt{4 - x^{2}} \, \mathrm{d}x$$ Deduce that $$m_{k} = \begin{cases} C_{k/2} & \text{if } k \text{ is even} \\ 0 & \text{if } k \text{ is odd.} \end{cases}$$
Q21 Measures of Location and Spread View
We assume that the random variables $X_{ij}$ are uniformly bounded: $\exists K \in \mathbb{R} \; \forall (i,j) \in (\mathbb{N}^{\star})^{2} \; |X_{ij}| \leqslant K$.
Let $k$ be a natural integer. Justify that the random variable $\sum_{i=1}^{n} \Lambda_{i,n}^{k}$ admits an expectation and that $$\mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} \Lambda_{i,n}^{k}\right) = \frac{1}{n^{1+k/2}} \mathbb{E}\left(\operatorname{tr}\left(M_{n}^{k}\right)\right) = \frac{1}{n^{1+k/2}} \sum_{(i_{1},\ldots,i_{k}) \in \llbracket 1,n \rrbracket^{k}} \mathbb{E}\left(X_{i_{1}i_{2}} X_{i_{2}i_{3}} \cdots X_{i_{k-1}i_{k}} X_{i_{k}i_{1}}\right).$$
Q22 Number Theory Combinatorial Number Theory and Counting View
We call a cycle of length $k$ with values in $\llbracket 1,n \rrbracket$, any $(k+1)$-tuple $\vec{\imath} = (i_{1}, i_{2}, \ldots, i_{k}, i_{1})$ of elements of $\llbracket 1,n \rrbracket$. We denote $|\vec{\imath}|$ the number of distinct vertices of the cycle $\vec{\imath}$.
Show that the number of cycles of length $k$ in $\llbracket 1,n \rrbracket$ passing through $\ell$ distinct vertices is at most $n^{\ell} \ell^{k}$.
Q23 Sequences and Series Limit Evaluation Involving Sequences View
We call a cycle of length $k$ with values in $\llbracket 1,n \rrbracket$, any $(k+1)$-tuple $\vec{\imath} = (i_{1}, i_{2}, \ldots, i_{k}, i_{1})$ of elements of $\llbracket 1,n \rrbracket$. We denote $|\vec{\imath}|$ the number of distinct vertices of the cycle $\vec{\imath}$.
Deduce that $$\frac{1}{n^{1+k/2}} \sum_{\substack{\vec{\imath} \in \llbracket 1,n \rrbracket^{k} \\ |\vec{\imath}| \leqslant (k+1)/2}} \left|\mathbb{E}\left(X_{i_{1}i_{2}} X_{i_{2}i_{3}} \cdots X_{i_{k-1}i_{k}} X_{i_{k}i_{1}}\right)\right| \xrightarrow{n \rightarrow +\infty} 0.$$
Q24 Sequences and Series Functional Equations and Identities via Series View
We classify cycles of length $k$ into three subsets:
  • the set $\mathcal{A}_{k}$, consisting of cycles where at least one edge appears only once;
  • the set $\mathcal{B}_{k}$, consisting of cycles where all edges appear exactly twice;
  • the set $\mathcal{C}_{k}$, consisting of cycles where all edges appear at least twice and there exists at least one that appears at least three times.

Show that, if the cycle $(i_{1}, i_{2}, \ldots, i_{k}, i_{1})$ belongs to $\mathcal{A}_{k}$, then $$\mathbb{E}\left(X_{i_{1}i_{2}} X_{i_{2}i_{3}} \cdots X_{i_{k-1}i_{k}} X_{i_{k}i_{1}}\right) = 0.$$
Q25 Number Theory Combinatorial Number Theory and Counting View
We classify cycles of length $k$ into three subsets:
  • the set $\mathcal{A}_{k}$, consisting of cycles where at least one edge appears only once;
  • the set $\mathcal{B}_{k}$, consisting of cycles where all edges appear exactly twice;
  • the set $\mathcal{C}_{k}$, consisting of cycles where all edges appear at least twice and there exists at least one that appears at least three times.

Show that, for every cycle $\vec{\imath}$ belonging to $\mathcal{C}_{k}$, $|\vec{\imath}| \leqslant \frac{k+1}{2}$.
Q28 Permutations & Arrangements Combinatorial Proof or Identity Derivation View
Assume that $k$ is even and that $\vec{\imath} \in \mathcal{B}_{k}$ is a cycle passing through $\frac{k}{2} + 1$ distinct vertices (i.e. $|\vec{\imath}| = \frac{k}{2} + 1$). We traverse the edges of $\vec{\imath}$ in order. To each edge of $\vec{\imath}$ we associate an opening parenthesis if this edge appears for the first time and a closing parenthesis if it appears for the second time.
Count the cycles $\vec{\imath}$ that correspond to a fixed well-parenthesized word.
Q30 Sequences and Series Evaluation of a Finite or Infinite Sum View
Deduce that, for every polynomial $P \in \mathbb{R}[X]$, $$\lim_{n \rightarrow +\infty} \mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} P\left(\Lambda_{i,n}\right)\right) = \frac{1}{2\pi} \int_{-2}^{2} P(x) \sqrt{4 - x^{2}} \, \mathrm{d}x$$
Q31 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View
Let $A > 2$. Show that, for every $(p,q) \in \mathbb{N}^{2}$, $$\mathbb{E}\left(\sum_{\substack{1 \leqslant i \leqslant n \\ |\Lambda_{i,n}| \geqslant A}} |\Lambda_{i,n}|^{p}\right) \leqslant \frac{1}{A^{p+2q}} \mathbb{E}\left(\sum_{i=1}^{n} |\Lambda_{i,n}|^{2(p+q)}\right)$$
Q32 Sequences and Series Limit Evaluation Involving Sequences View
Let $A > 2$. Show that $$\lim_{n \rightarrow +\infty} \frac{1}{n} \mathbb{E}\left(\sum_{\substack{1 \leqslant i \leqslant n \\ |\Lambda_{i,n}| \geqslant A}} |\Lambda_{i,n}|^{p}\right) = 0.$$
Q33 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $A > 2$. Let $f$ be a continuous and bounded function from $\mathbb{R}$ to $\mathbb{R}$ and $P$ a polynomial of degree $p$. Justify that there exists a constant $K$ such that $$\forall x \in \mathbb{R} \setminus ]-A, A[, \quad |f(x) - P(x)| \leqslant K|x|^{p}.$$
Q34 Continuous Probability Distributions and Random Variables Almost Sure Convergence and Random Series Properties View
Let $A > 2$. Deduce that $$\lim_{n \rightarrow +\infty} \frac{1}{n} \mathbb{E}\left(\sum_{\substack{1 \leqslant i \leqslant n \\ |\Lambda_{i,n}| \geqslant A}} |f - P|\left(\Lambda_{i,n}\right)\right) = 0.$$
Q35 Continuous Probability Distributions and Random Variables Convergence in Distribution or Probability View
Let $f$ be a continuous and bounded function from $\mathbb{R}$ to $\mathbb{R}$. Show that $$\lim_{n \rightarrow +\infty} \mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} f\left(\Lambda_{i,n}\right)\right) = \frac{1}{2\pi} \int_{-2}^{2} f(x) \sqrt{4 - x^{2}} \, \mathrm{d}x$$
Q36 Discrete Random Variables Dominated Convergence and Truncation Arguments for Discrete Variables View
Let $X$ be a discrete random variable with finite expectation. Show that $$\mathbb{E}\left(X \mathbb{1}_{|X| \leqslant C}\right) \xrightarrow{C \rightarrow +\infty} \mathbb{E}(X).$$
Q37 Discrete Random Variables Dominated Convergence and Truncation Arguments for Discrete Variables View
For every $(i,j) \in (\mathbb{N}^{\star})^{2}$ and for every $C > 0$, we set $\sigma_{ij}(C) = \sqrt{\mathbb{V}\left(X_{ij} \mathbb{1}_{|X_{ij}| \leqslant C}\right)}$.
Deduce that $$\lim_{C \rightarrow +\infty} \sigma_{ij}(C) = 1$$
Q38 Discrete Random Variables Dominated Convergence and Truncation Arguments for Discrete Variables View
For every $(i,j) \in (\mathbb{N}^{\star})^{2}$ and for every $C > 0$, we set $\sigma_{ij}(C) = \sqrt{\mathbb{V}\left(X_{ij} \mathbb{1}_{|X_{ij}| \leqslant C}\right)}$. If $\sigma_{ij}(C) \neq 0$, we set $$\widehat{X}_{ij}(C) = \frac{1}{\sigma_{ij}(C)}\left(X_{ij} \mathbb{1}_{|X_{ij}| \leqslant C} - \mathbb{E}\left(X_{ij} \mathbb{1}_{|X_{ij}| \leqslant C}\right)\right).$$
Justify that, for $C$ large enough, the variables $\widehat{X}_{ij}(C)$ are well defined and that they are then bounded, centered, of variance 1 and that they are mutually independent for $1 \leqslant i \leqslant j$.
Q39 Discrete Random Variables Dominated Convergence and Truncation Arguments for Discrete Variables View
For every $(i,j) \in (\mathbb{N}^{\star})^{2}$ and for every $C > 0$, we set $\sigma_{ij}(C) = \sqrt{\mathbb{V}\left(X_{ij} \mathbb{1}_{|X_{ij}| \leqslant C}\right)}$. If $\sigma_{ij}(C) \neq 0$, we set $$\widehat{X}_{ij}(C) = \frac{1}{\sigma_{ij}(C)}\left(X_{ij} \mathbb{1}_{|X_{ij}| \leqslant C} - \mathbb{E}\left(X_{ij} \mathbb{1}_{|X_{ij}| \leqslant C}\right)\right).$$
Show that $$X_{ij} - \widehat{X}_{ij}(C) = \left(1 - \frac{1}{\sigma_{ij}(C)}\right) X_{ij} + \frac{1}{\sigma_{ij}(C)}\left(X_{ij} \mathbb{1}_{|X_{ij}| > C} - \mathbb{E}\left(X_{ij} \mathbb{1}_{|X_{ij}| > C}\right)\right).$$
Q40 Discrete Random Variables Dominated Convergence and Truncation Arguments for Discrete Variables View
For every $(i,j) \in (\mathbb{N}^{\star})^{2}$ and for every $C > 0$, we set $\sigma_{ij}(C) = \sqrt{\mathbb{V}\left(X_{ij} \mathbb{1}_{|X_{ij}| \leqslant C}\right)}$. If $\sigma_{ij}(C) \neq 0$, we set $$\widehat{X}_{ij}(C) = \frac{1}{\sigma_{ij}(C)}\left(X_{ij} \mathbb{1}_{|X_{ij}| \leqslant C} - \mathbb{E}\left(X_{ij} \mathbb{1}_{|X_{ij}| \leqslant C}\right)\right).$$
Show that $$\lim_{C \rightarrow +\infty} \mathbb{E}\left(\left(X_{ij} - \widehat{X}_{ij}(C)\right)^{2}\right) = 0.$$
Q41 Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables View
For every integer $n$ such that $n \geqslant 1$, we denote by $\widehat{M}_{n}(C) = \left(\widehat{X}_{ij}(C)\right)_{1 \leqslant i,j \leqslant n}$. For every $\omega \in \Omega$, we denote by $\widehat{\Lambda}_{1,n}(\omega) \geqslant \ldots \geqslant \widehat{\Lambda}_{n,n}(\omega)$ the eigenvalues of $\frac{1}{\sqrt{n}} \widehat{M}_{n}(C)(\omega)$ arranged in decreasing order. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a $K$-Lipschitz function.
Show that $$\left|\mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} f\left(\Lambda_{i,n}\right)\right) - \mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} f\left(\widehat{\Lambda}_{i,n}\right)\right)\right| \leqslant \frac{K}{n} \mathbb{E}\left(\left\|M_{n} - \widehat{M}_{n}(C)\right\|_{F}\right).$$
Q42 Discrete Random Variables Convergence to a Limiting Distribution (Semicircle Law and Analogues) View
For every integer $n$ such that $n \geqslant 1$, we denote by $\widehat{M}_{n}(C) = \left(\widehat{X}_{ij}(C)\right)_{1 \leqslant i,j \leqslant n}$. For every $\omega \in \Omega$, we denote by $\widehat{\Lambda}_{1,n}(\omega) \geqslant \ldots \geqslant \widehat{\Lambda}_{n,n}(\omega)$ the eigenvalues of $\frac{1}{\sqrt{n}} \widehat{M}_{n}(C)(\omega)$ arranged in decreasing order. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a $K$-Lipschitz function.
Assume furthermore that $f$ is bounded. Show $$\mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} f\left(\Lambda_{i,n}\right)\right) \xrightarrow{n \rightarrow +\infty} \frac{1}{2\pi} \int_{-2}^{2} f(x) \sqrt{4 - x^{2}} \, \mathrm{d}x.$$
Q43 Discrete Random Variables Convergence to a Limiting Distribution (Semicircle Law and Analogues) View
Show the semicircle law in the general case:
For every function $f : \mathbb{R} \rightarrow \mathbb{R}$, continuous and bounded, $$\mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} f\left(\Lambda_{i,n}\right)\right) \xrightarrow{n \rightarrow +\infty} \frac{1}{2\pi} \int_{-2}^{2} f(x) \sqrt{4 - x^{2}} \, \mathrm{d}x$$