grandes-ecoles

Q1 Proof Direct Proof of an Inequality View

Show the interpolation inequality $$\forall f \in \mathcal{C}^1([0,1]), \quad \|f\|_\infty \leqslant \left\|f^\prime\right\|_\infty + C\left|f\left(x_1\right)\right|$$ with $C = 1$.

Q3 Proof Direct Proof of an Inequality View

We fix two distinct real numbers $x_1 < x_2$ in $[0,1]$. For all $x \in [0,1]$ and $f \in \mathcal{C}^2([0,1])$, prove the inequality $$\left|f^\prime(x) - \frac{f\left(x_2\right) - f\left(x_1\right)}{x_2 - x_1}\right| \leqslant \left\|f^{\prime\prime}\right\|_\infty.$$

Q5 Proof Direct Proof of an Inequality View

We fix two distinct real numbers $x_1 < x_2$ in $[0,1]$. Conclude the case $K = 2$ by showing the interpolation inequality $$\forall f \in \mathcal{C}^2([0,1]), \quad \max\left(\|f\|_\infty, \left\|f^\prime\right\|_\infty\right) \leqslant \left\|f^{\prime\prime}\right\|_\infty + C\left(\left|f\left(x_1\right)\right| + \left|f\left(x_2\right)\right|\right)$$ with $C = 1 + \frac{1}{x_2 - x_1}$.

Q6 Proof Proof That a Map Has a Specific Property View

Prove that the map $$\begin{array}{ccl} \Psi : \mathbb{R}_{K-1}[X] & \rightarrow & \mathbb{R}^K \\ P & \mapsto & \left(P\left(x_1\right), \ldots, P\left(x_K\right)\right) \end{array}$$ is an isomorphism of vector spaces.

Q7 Factor & Remainder Theorem Lagrange Interpolation and Basis Representation View

Show that there exist $K$ polynomials $L_1, \ldots, L_K$ in $\mathbb{R}_{K-1}[X]$ such that, for any function $f \in \mathcal{C}^K([0,1])$, the polynomial $P = \sum_{j=1}^K f\left(x_j\right) L_j$ satisfies $$\forall \ell \in \llbracket 1, K \rrbracket, \quad P\left(x_\ell\right) = f\left(x_\ell\right).$$

Q8 Proof Existence Proof View

We fix $f \in \mathcal{C}^K([0,1])$ and denote by $P$ the polynomial determined in question Q7. For all $k \in \llbracket 0, K-1 \rrbracket$, show that there exist at least $K - k$ distinct real numbers in $[0,1]$ at which the function $f^{(k)} - P^{(k)}$ vanishes.

Q9 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

We fix $f \in \mathcal{C}^K([0,1])$ and denote by $P$ the polynomial determined in question Q7. Deduce the inequality $\left\|f^{(k)} - P^{(k)}\right\|_\infty \leqslant \left\|f^{(k+1)} - P^{(k+1)}\right\|_\infty$ for all $k \in \llbracket 0, K-1 \rrbracket$.

Q10 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Show that there exists a constant $C > 0$ for which the interpolation inequality $$\forall f \in \mathcal{C}^K([0,1]), \quad \max_{0 \leqslant k \leqslant K-1} \left\|f^{(k)}\right\|_\infty \leqslant \left\|f^{(K)}\right\|_\infty + C \sum_{\ell=1}^K \left|f\left(x_\ell\right)\right|$$ is satisfied.

Q11 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View

Let $K \in \mathbb{N}^\star$, consider distinct real numbers $x_1 < \cdots < x_K$ in an interval $[a,b]$ (with $a < b$), and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[a,b]$ with real values satisfying: (H1) the function series $\sum f_n^{(K)}$ converges normally on $[a,b]$; (H2) for all $\ell \in \llbracket 1, K \rrbracket$ the numerical series $\sum f_n(x_\ell)$ is absolutely convergent.
In the particular case $[a,b] = [0,1]$, justify that the series $\sum f_n^{(k)}$ converges normally on $[a,b]$ for all $k \in \llbracket 0, K-1 \rrbracket$.

Q12 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View

Let $K \in \mathbb{N}^\star$, consider distinct real numbers $x_1 < \cdots < x_K$ in an interval $[a,b]$ (with $a < b$), and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[a,b]$ with real values satisfying: (H1) the function series $\sum f_n^{(K)}$ converges normally on $[a,b]$; (H2) for all $\ell \in \llbracket 1, K \rrbracket$ the numerical series $\sum f_n(x_\ell)$ is absolutely convergent.
Treat the previous question in the general case of a segment $[a,b]$ with $a < b$. One may examine $f_n \circ \sigma$ where $\sigma : [0,1] \rightarrow [a,b]$ is defined by $\sigma(t) = (1-t)a + tb$ for all $t \in [0,1]$.

Q13 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View

Let $K \in \mathbb{N}^\star$, consider distinct real numbers $x_1 < \cdots < x_K$ in an interval $[a,b]$ (with $a < b$), and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[a,b]$ with real values satisfying: (H1) the function series $\sum f_n^{(K)}$ converges normally on $[a,b]$; (H2) for all $\ell \in \llbracket 1, K \rrbracket$ the numerical series $\sum f_n(x_\ell)$ is absolutely convergent.
According to the result of the previous question, we can set $F_k(x) = \sum_{n=0}^{+\infty} f_n^{(k)}(x)$ for all $x \in [a,b]$. Prove that $F_0$ is of class $\mathcal{C}^K$ on $[a,b]$ and that $F_0^{(k)} = F_k$ for all $k \in \llbracket 1, K \rrbracket$.

Q14 Differential equations Higher-Order and Special DEs (Proof/Theory) View

For all $n \in \mathbb{N}^\star$, justify that there exists a unique function $f_n \in \mathcal{C}^2(]0,+\infty[)$ satisfying $f_n(1) = 0$, $f_n(2) = 0$ and $f_n^{\prime\prime}(x) = (-1)^n 2^{-nx^2}$ for all $x > 0$.

Q15 Proof Proof That a Map Has a Specific Property View

For all $n \in \mathbb{N}^\star$, let $f_n \in \mathcal{C}^2(]0,+\infty[)$ satisfy $f_n(1) = 0$, $f_n(2) = 0$ and $f_n^{\prime\prime}(x) = (-1)^n 2^{-nx^2}$ for all $x > 0$. Show that the function series $\sum f_n(x)$ converges normally on any segment contained in $]0,+\infty[$ and that the function $F : x \mapsto \sum_{n=1}^{+\infty} f_n(x)$ is of class $\mathcal{C}^2$ on $]0,+\infty[$.

Q16 Sequences and Series Evaluation of a Finite or Infinite Sum View

For all $n \in \mathbb{N}^\star$, let $f_n \in \mathcal{C}^2(]0,+\infty[)$ satisfy $f_n(1) = 0$, $f_n(2) = 0$ and $f_n^{\prime\prime}(x) = (-1)^n 2^{-nx^2}$ for all $x > 0$. Let $F : x \mapsto \sum_{n=1}^{+\infty} f_n(x)$. Explicitly state $F^{\prime\prime}(x)$.

Q17 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

For all $n \in \mathbb{N}^\star$, let $f_n \in \mathcal{C}^2(]0,+\infty[)$ satisfy $f_n(1) = 0$, $f_n(2) = 0$ and $f_n^{\prime\prime}(x) = (-1)^n 2^{-nx^2}$ for all $x > 0$. Let $F : x \mapsto \sum_{n=1}^{+\infty} f_n(x)$. Show that $|F(x)| \leqslant \frac{1}{3}$ for all $x \in [1,2]$.

Q18 Sequences and series, recurrence and convergence Series convergence and power series analysis View

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, and let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that the series $\sum a_n^2$ converges. Justify the existence of a strictly increasing sequence of natural integers $(\phi(j))_{j \in \mathbb{N}}$ satisfying $$\forall j \in \mathbb{N}, \quad \sum_{n > \phi(j)}^{+\infty} a_n^2 \leqslant \frac{1}{8^j}.$$

Q19 Discrete Random Variables Expectation and Variance of Sums of Independent Variables View

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, and let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that the series $\sum a_n^2$ converges. For all $N \in \mathbb{N}$, denote $S_N = \sum_{n=0}^N X_n a_n$. Let $(\phi(j))_{j \in \mathbb{N}}$ be a strictly increasing sequence of natural integers satisfying $\sum_{n > \phi(j)}^{+\infty} a_n^2 \leqslant \frac{1}{8^j}$ for all $j \in \mathbb{N}$. Express the expectation and variance of $S_{\phi(j+1)} - S_{\phi(j)}$ in terms of the terms of the sequence $(a_n)_{n \in \mathbb{N}}$.

Q20 Continuous Probability Distributions and Random Variables Probability Inequality and Tail Bound Proof View

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, and let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that the series $\sum a_n^2$ converges. For all $N \in \mathbb{N}$, denote $S_N = \sum_{n=0}^N X_n a_n$. Let $(\phi(j))_{j \in \mathbb{N}}$ be a strictly increasing sequence of natural integers satisfying $\sum_{n > \phi(j)}^{+\infty} a_n^2 \leqslant \frac{1}{8^j}$ for all $j \in \mathbb{N}$, and let $A_j = \left\{\left|S_{\phi(j+1)} - S_{\phi(j)}\right| > 2^{-j}\right\}$. Deduce from the two previous questions the bound $\mathbb{P}(A_j) \leqslant 2^{-j}$.

Q21 Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables View

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, and let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that the series $\sum a_n^2$ converges. For all $N \in \mathbb{N}$, denote $S_N = \sum_{n=0}^N X_n a_n$. Let $(\phi(j))_{j \in \mathbb{N}}$ be a strictly increasing sequence of natural integers. Define the events $$B_j = \left\{\max_{\phi(j)+1 \leqslant n \leqslant \phi(j+1)} \left|S_n - S_{\phi(j)}\right| > 2^{-j}\right\},$$ $$B_{j,m} = \left\{\left|S_m - S_{\phi(j)}\right| > 2^{-j} \text{ and } \forall n \in \llbracket \phi(j), m-1 \rrbracket, \quad \left|S_n - S_{\phi(j)}\right| \leqslant 2^{-j}\right\}.$$ For all $j \in \mathbb{N}$, prove that the events $B_{j,m}$, for $m$ ranging over $\llbracket \phi(j)+1, \phi(j+1) \rrbracket$, are pairwise disjoint and that we have the equality of events $$B_j = \bigcup_{\phi(j) < m \leqslant \phi(j+1)} B_{j,m}.$$

Q22 Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables View

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, and let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that the series $\sum a_n^2$ converges. For all $N \in \mathbb{N}$, denote $S_N = \sum_{n=0}^N X_n a_n$. Let $(\phi(j))_{j \in \mathbb{N}}$ be a strictly increasing sequence of natural integers. Define the events $$A_j = \left\{\left|S_{\phi(j+1)} - S_{\phi(j)}\right| > 2^{-j}\right\},$$ $$B_{j,m} = \left\{\left|S_m - S_{\phi(j)}\right| > 2^{-j} \text{ and } \forall n \in \llbracket \phi(j), m-1 \rrbracket, \quad \left|S_n - S_{\phi(j)}\right| \leqslant 2^{-j}\right\}.$$ Explain how to deduce the formula $\mathbb{P}(A_j) = \sum_{m=\phi(j)+1}^{\phi(j+1)} \mathbb{P}(A_j \cap B_{j,m})$.

Q23 Continuous Probability Distributions and Random Variables Almost Sure Convergence and Random Series Properties View

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, and let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that the series $\sum a_n^2$ converges. For all $N \in \mathbb{N}$, denote $S_N = \sum_{n=0}^N X_n a_n$. Let $(\phi(j))_{j \in \mathbb{N}}$ be a strictly increasing sequence of natural integers. Define $$B_{j,m} = \left\{\left|S_m - S_{\phi(j)}\right| > 2^{-j} \text{ and } \forall n \in \llbracket \phi(j), m-1 \rrbracket, \quad \left|S_n - S_{\phi(j)}\right| \leqslant 2^{-j}\right\}.$$ Let $m \in \llbracket \phi(j)+1, \phi(j+1) \rrbracket$, show that the function $$\left|\begin{array}{rcl} \mathbb{R} & \rightarrow & \mathbb{R} \\ \alpha & \mapsto & 2^{\phi(j+1)-\phi(j)} \mathbb{P}\left(\left\{\left|\alpha S_{\phi(j+1)} - \alpha S_m + S_m - S_{\phi(j)}\right| > 2^{-j}\right\} \cap B_{j,m}\right) \end{array}\right.$$ takes values in $\mathbb{N}$ and is even.

Q24 Continuous Probability Distributions and Random Variables Almost Sure Convergence and Random Series Properties View

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, and let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that the series $\sum a_n^2$ converges. For all $N \in \mathbb{N}$, denote $S_N = \sum_{n=0}^N X_n a_n$. Let $(\phi(j))_{j \in \mathbb{N}}$ be a strictly increasing sequence of natural integers. Define $$B_j = \left\{\max_{\phi(j)+1 \leqslant n \leqslant \phi(j+1)} \left|S_n - S_{\phi(j)}\right| > 2^{-j}\right\},$$ $$B_{j,m} = \left\{\left|S_m - S_{\phi(j)}\right| > 2^{-j} \text{ and } \forall n \in \llbracket \phi(j), m-1 \rrbracket, \quad \left|S_n - S_{\phi(j)}\right| \leqslant 2^{-j}\right\}.$$ Prove that if the event $B_j$ occurs, then there exist $m \in \llbracket \phi(j)+1, \phi(j+1) \rrbracket$ and $\alpha \in \{-1,+1\}$ such that the event $$\left\{\left|\alpha S_{\phi(j+1)} - \alpha S_m + S_m - S_{\phi(j)}\right| > 2^{-j}\right\} \cap B_{j,m}$$ also occurs. One may express $S_m - S_{\phi(j)}$ in terms of the two numbers $\alpha S_{\phi(j+1)} - \alpha S_m + S_m - S_{\phi(j)}$ with $\alpha = \pm 1$.

Q25 Continuous Probability Distributions and Random Variables Probability Inequality and Tail Bound Proof View

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, and let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that the series $\sum a_n^2$ converges. For all $N \in \mathbb{N}$, denote $S_N = \sum_{n=0}^N X_n a_n$. Let $(\phi(j))_{j \in \mathbb{N}}$ be a strictly increasing sequence of natural integers. Define $$A_j = \left\{\left|S_{\phi(j+1)} - S_{\phi(j)}\right| > 2^{-j}\right\}, \quad B_j = \left\{\max_{\phi(j)+1 \leqslant n \leqslant \phi(j+1)} \left|S_n - S_{\phi(j)}\right| > 2^{-j}\right\}.$$ Deduce that $\mathbb{P}(B_j) \leqslant 2\mathbb{P}(A_j)$.

Q26 Continuous Probability Distributions and Random Variables Almost Sure Convergence and Random Series Properties View

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, and let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that the series $\sum a_n^2$ converges. For all $N \in \mathbb{N}$, denote $S_N = \sum_{n=0}^N X_n a_n$. Let $(\phi(j))_{j \in \mathbb{N}}$ be a strictly increasing sequence of natural integers satisfying $\sum_{n > \phi(j)}^{+\infty} a_n^2 \leqslant \frac{1}{8^j}$ for all $j \in \mathbb{N}$. Define $B_j = \left\{\max_{\phi(j)+1 \leqslant n \leqslant \phi(j+1)} \left|S_n - S_{\phi(j)}\right| > 2^{-j}\right\}$. We denote by $B$ the event $\bigcap_{J \in \mathbb{N}} \bigcup_{j \geqslant J} B_j$. Show the equality $\mathbb{P}(B) = 0$.

Q27 Continuous Probability Distributions and Random Variables Almost Sure Convergence and Random Series Properties View

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, and let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that the series $\sum a_n^2$ converges. For all $N \in \mathbb{N}$, denote $S_N = \sum_{n=0}^N X_n a_n$. Let $(\phi(j))_{j \in \mathbb{N}}$ be a strictly increasing sequence of natural integers satisfying $\sum_{n > \phi(j)}^{+\infty} a_n^2 \leqslant \frac{1}{8^j}$ for all $j \in \mathbb{N}$. Show that the event $$\left\{\exists J \in \mathbb{N}, \quad \forall j \geqslant J, \quad \forall n \in \llbracket \phi(j)+1, \phi(j+1) \rrbracket, \quad \left|S_n - S_{\phi(j)}\right| \leqslant 2^{-j}\right\}$$ occurs with probability 1.

Q28 Continuous Probability Distributions and Random Variables Almost Sure Convergence and Random Series Properties View

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, and let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that the series $\sum a_n^2$ converges. For all $N \in \mathbb{N}$, denote $S_N = \sum_{n=0}^N X_n a_n$. Let $(\phi(j))_{j \in \mathbb{N}}$ be a strictly increasing sequence of natural integers satisfying $\sum_{n > \phi(j)}^{+\infty} a_n^2 \leqslant \frac{1}{8^j}$ for all $j \in \mathbb{N}$. Deduce that the event $$\left\{\text{the sequence } \left(S_{\phi(j)}\right)_{j \in \mathbb{N}} \text{ is convergent}\right\}$$ also has probability 1. One may examine the series $\sum \left|S_{\phi(j+1)} - S_{\phi(j)}\right|$.

Q29 Continuous Probability Distributions and Random Variables Almost Sure Convergence and Random Series Properties View

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, and let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that the series $\sum a_n^2$ converges. For all $N \in \mathbb{N}$, denote $S_N = \sum_{n=0}^N X_n a_n$. Let $(\phi(j))_{j \in \mathbb{N}}$ be a strictly increasing sequence of natural integers satisfying $\sum_{n > \phi(j)}^{+\infty} a_n^2 \leqslant \frac{1}{8^j}$ for all $j \in \mathbb{N}$. Conclude that the event $$\left\{\text{the series } \sum X_n a_n \text{ is convergent}\right\}$$ has probability 1.

Q30 Continuous Probability Distributions and Random Variables Almost Sure Convergence and Random Series Properties View

We fix $K \in \mathbb{N}^\star$ and consider a sequence of random variables $(X_n)_{n \in \mathbb{N}}$ satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, distinct real numbers $x_1 < \cdots < x_K$ in $[0,1]$, and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[0,1]$ with real values satisfying: (H1) the function series $\sum f_n^{(K)}$ converges normally on $[0,1]$; (H2') for all $\ell \in \llbracket 1, K \rrbracket$, the numerical series $\sum f_n(x_\ell)^2$ is convergent.
Show that one of the two hypotheses (H2') or (H2) (studied in part II, where (H2) states that for all $\ell \in \llbracket 1, K \rrbracket$ the numerical series $\sum f_n(x_\ell)$ is absolutely convergent) implies the other.

Q31 Continuous Probability Distributions and Random Variables Almost Sure Convergence and Random Series Properties View

We fix $K \in \mathbb{N}^\star$ and consider a sequence of random variables $(X_n)_{n \in \mathbb{N}}$ satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, distinct real numbers $x_1 < \cdots < x_K$ in $[0,1]$, and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[0,1]$ with real values satisfying: (H1) the function series $\sum f_n^{(K)}$ converges normally on $[0,1]$; (H2') for all $\ell \in \llbracket 1, K \rrbracket$, the numerical series $\sum f_n(x_\ell)^2$ is convergent.
Show that the event $$\left\{\text{for all } \ell \in \llbracket 1, K \rrbracket, \text{ the series } \sum X_n f_n(x_\ell) \text{ is convergent}\right\}$$ has probability 1.

Q32 Continuous Probability Distributions and Random Variables Almost Sure Convergence and Random Series Properties View

We fix $K \in \mathbb{N}^\star$ and consider a sequence of random variables $(X_n)_{n \in \mathbb{N}}$ satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, distinct real numbers $x_1 < \cdots < x_K$ in $[0,1]$, and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[0,1]$ with real values satisfying: (H1) the function series $\sum f_n^{(K)}$ converges normally on $[0,1]$; (H2') for all $\ell \in \llbracket 1, K \rrbracket$, the numerical series $\sum f_n(x_\ell)^2$ is convergent.
Let $P_n \in \mathbb{R}_{K-1}[X]$ be a polynomial satisfying $P_n(x_\ell) = f_n(x_\ell)$ for all $\ell \in \llbracket 1, K \rrbracket$ (cf. question 7). Show that the event $$\left\{\begin{array}{l} \text{for all } k \in \llbracket 0, K \rrbracket, \text{ the function series } \sum X_n (f_n - P_n)^{(k)} \text{ is uniformly convergent on } [0,1], \\ \text{the function } \sum_{n=0}^{+\infty} X_n (f_n - P_n) \text{ is of class } \mathcal{C}^K, \\ \text{for all } k \in \llbracket 0, K \rrbracket, \left(\sum_{n=0}^{+\infty} X_n (f_n - P_n)\right)^{(k)} = \sum_{n=0}^{+\infty} X_n (f_n - P_n)^{(k)} \end{array}\right\}$$ has probability 1.

Q33 Continuous Probability Distributions and Random Variables Almost Sure Convergence and Random Series Properties View

We fix $K \in \mathbb{N}^\star$ and consider a sequence of random variables $(X_n)_{n \in \mathbb{N}}$ satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, distinct real numbers $x_1 < \cdots < x_K$ in $[0,1]$, and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[0,1]$ with real values satisfying: (H1) the function series $\sum f_n^{(K)}$ converges normally on $[0,1]$; (H2') for all $\ell \in \llbracket 1, K \rrbracket$, the numerical series $\sum f_n(x_\ell)^2$ is convergent.
Show that the event $$\left\{\begin{array}{l} \text{for all } k \in \llbracket 0, K \rrbracket, \text{ the function series } \sum X_n f_n^{(k)} \text{ is uniformly convergent on } [0,1], \\ \text{the function } \sum_{n=0}^{+\infty} X_n f_n \text{ is of class } \mathcal{C}^K, \\ \text{for all } k \in \llbracket 0, K \rrbracket, \left(\sum_{n=0}^{+\infty} X_n f_n\right)^{(k)} = \sum_{n=0}^{+\infty} X_n f_n^{(k)} \end{array}\right\}$$ has probability 1.

Q34 Continuous Probability Distributions and Random Variables Almost Sure Convergence and Random Series Properties View

Give an example of an integer $K \in \mathbb{N}^\star$ for which the event $$\left\{\begin{array}{l} \text{for all } k \in \llbracket 0, K \rrbracket, \text{ the function series } \sum X_n f_n^{(k)} \text{ is uniformly convergent on } [0,1], \\ \text{the function } \sum_{n=0}^{+\infty} X_n f_n \text{ is of class } \mathcal{C}^K, \\ \text{for all } k \in \llbracket 0, K \rrbracket, \left(\sum_{n=0}^{+\infty} X_n f_n\right)^{(k)} = \sum_{n=0}^{+\infty} X_n f_n^{(k)} \end{array}\right\}$$ occurs with the functions $f_n$ defined by $$\left\{\begin{array}{l} f_0 = 0 \\ f_n(x) = \ln\left(1 + \sin\left(\frac{x}{n}\right)\right) \quad \forall n \in \mathbb{N}^\star, \forall x \in [0,1]. \end{array}\right.$$

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