Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \; A \in \mathscr{P}(E)\}$. Deduce that the sequence $(\mu_n)_{n \in \mathbb{N}}$ converges to $\mu$ in $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ if and only if it satisfies condition (1): $\forall x \in E, \lim_{n \to +\infty} \mu_n(x) = \mu(x)$.

For all functions $f \in E$ and $g \in E$, we set $\langle f \mid g \rangle = \int _ { 0 } ^ { + \infty } f ( t ) g ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Show that this defines an inner product on $E$.

Let $\mathcal{C}^{1}$ be the space of functions of class $C^{1}$ from $[-\pi, \pi]$ to $\mathbf{C}$. For $f \in \mathcal{C}^{1}$, we set $$\|f\|_{\infty} = \max\{|f(t)|; t \in [-\pi, \pi]\} \quad \text{and} \quad V(f) = \int_{-\pi}^{\pi} |f^{\prime}|.$$
By considering a well-chosen sequence of functions, show that there does not exist an element $C$ of $\mathbf{R}^{+*}$ such that $$\forall f \in \mathcal{C}^{1}, \quad V(f) \leq C\|f\|_{\infty}$$

Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be a countably infinite set where the $x_i$ are pairwise distinct elements. We denote by $\mathscr{M}(E)$ the set of probability measures on $E$. We denote by $\mathscr{P}(E)$ the set of subsets of $E$. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}$.
For every integer $k \in \mathbb{N}^*$, we denote by $\delta_k$ the probability measure on $E$ such that, for all $n \in \mathbb{N}^*$, $$\delta_k(\{x_n\}) = \begin{cases} 1 & \text{if } n = k \\ 0 & \text{otherwise} \end{cases}.$$
Does the sequence $(\delta_k)_{k \in \mathbb{N}^*}$ converge in $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$?

Let $E = \{x_1, x_2, \ldots, x_n, \ldots\}$ be an infinite countable set. We denote $\mathscr{M}(E)$ the set of probability measures on $E$. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \; A \in \mathscr{P}(E)\}$. For every integer $k \in \mathbb{N}^*$, we denote $\delta_k$ the probability measure on $E$ such that, for all $n \in \mathbb{N}^*$, $$\delta_k(\{x_n\}) = \begin{cases} 1 & \text{if } n = k \\ 0 & \text{otherwise} \end{cases}$$ Does the sequence $(\delta_k)_{k \in \mathbb{N}^*}$ converge in $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$?

For $k \in \mathbf { N } ^ { * }$ and $t \in \mathbf { R } _ { + }$, we set
$$u _ { k } ( t ) = \int _ { k/2 } ^ { ( k + 1 ) / 2 } \frac { t q ( u ) } { e ^ { t u } - 1 } \mathrm {~d} u \quad \text { if } t > 0 , \quad \text { and } \quad u _ { k } ( t ) = \int _ { k/2 } ^ { ( k + 1 ) / 2 } \frac { q ( u ) } { u } \mathrm {~d} u \quad \text { if } t = 0$$
where $q ( x ) = x - \lfloor x \rfloor - \frac { 1 } { 2 }$.
Show that $u _ { k }$ is continuous on $\mathbf { R } _ { + }$ for all $k \in \mathbf { N } ^ { * }$.

For $k \in \mathbf{N}^*$ and $t \in \mathbf{R}_+$, we set $$u_k(t) = \int_{k/2}^{(k+1)/2} \frac{tq(u)}{e^{tu}-1} \mathrm{du} \text{ if } t > 0 \text{, and } u_k(t) = \int_{k/2}^{(k+1)/2} \frac{q(u)}{u} \mathrm{du} \text{ if } t = 0.$$ Show that $u_k$ is continuous on $\mathbf{R}$, for all $k \in \mathbf{N}^*$.

The norm $\| \cdot \|$ associated with the inner product $\langle f \mid g \rangle = \int _ { 0 } ^ { + \infty } f ( t ) g ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ is defined for all functions $f \in E$ by $$\| f \| = \left( \int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t \right) ^ { 1 / 2 }$$ Show that $\lim _ { x \rightarrow 0 } \left\| k _ { x } \right\| = 0$. We recall that, for all $x > 0 , k _ { x } ( t ) = \mathrm { e } ^ { \min ( x , t ) } - 1$.

Let $\mathcal{C}^{1}$ be the space of functions of class $C^{1}$ from $[-\pi, \pi]$ to $\mathbf{C}$. For $f \in \mathcal{C}^{1}$, we set $$\|f\|_{\infty} = \max\{|f(t)|; t \in [-\pi, \pi]\} \quad \text{and} \quad V(f) = \int_{-\pi}^{\pi} |f^{\prime}|.$$
Let $f \in \mathcal{C}^{1}$ with real values. We assume that the set $C(f)$ of points in $]-\pi, \pi[$ where the function $f^{\prime}$ vanishes is finite. We denote by $\ell$ the cardinality of $C(f)$ and, if $\ell \geq 1$, we denote by $t_{1} < \cdots < t_{\ell}$ the elements of $C(f)$. We set $t_{0} = -\pi$ and $t_{\ell+1} = \pi$.
Show that $$V(f) = \sum_{j=0}^{\ell} \left|f\left(t_{j+1}\right) - f\left(t_{j}\right)\right|$$
For $0 \leq j \leq \ell$, let $\psi_{j}$ be the function from $\mathbf{R}$ to $\{0,1\}$ equal to 1 on $\left[f\left(t_{j}\right), f\left(t_{j+1}\right)\right[$ and to 0 on $\mathbf{R} \backslash \left[f\left(t_{j}\right), f\left(t_{j+1}\right)\right[$. Show that $$V(f) = \sum_{j=0}^{\ell} \int_{-\|f\|_{\infty}}^{\|f\|_{\infty}} \psi_{j}$$

Let $E$ be a countably infinite subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables defined on $(\Omega, \mathscr{A}, P)$, taking values in $E$. We assume that for all $\omega \in \Omega$, the sequence $(X_n(\omega))_{n \in \mathbb{N}}$ is stationary and converges to $X(\omega)$. We also define the random variable: $$L : \Omega \longrightarrow \mathbb{N}, \quad \omega \mapsto \begin{cases} 0 & \text{if } \forall n \in \mathbb{N}, X_n(\omega) = X(\omega) \\ \max\{n \in \mathbb{N}, X_n(\omega) \neq X(\omega)\} & \text{otherwise.} \end{cases}$$
Justify that the map $L$ is well defined.

Let $E$ be a countably infinite subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables defined on $(\Omega, \mathscr{A}, P)$, taking values in $E$. We assume that for all $\omega \in \Omega$, the sequence $(X_n(\omega))_{n \in \mathbb{N}}$ is stationary and converges to $X(\omega)$. Let $L$ be the random variable defined as in 15a.
Show that $P(X_n \neq X) \leqslant P(L \geqslant n)$ for all integer $n$ in $\mathbb{N}$.

Let $E$ be a countably infinite subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables defined on $(\Omega, \mathscr{A}, P)$, taking values in $E$. We assume that for all $\omega \in \Omega$, the sequence $(X_n(\omega))_{n \in \mathbb{N}}$ is stationary and converges to $X(\omega)$. Let $L$ be the random variable defined as in 15a.
Deduce that $\lim_{n \rightarrow +\infty} \|\mu_{X_n} - \mu_X\| = 0$.

Let $E$ be an infinite countable subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables defined on $(\Omega, \mathscr{A}, P)$, taking values in $E$, such that for all $\omega \in \Omega$, the sequence $(X_n(\omega))_{n \in \mathbb{N}}$ is stationary and converges to $X(\omega)$. We define the random variable: $$L : \Omega \longrightarrow \mathbb{N}, \quad \omega \longmapsto \begin{cases} 0 & \text{if } \forall n \in \mathbb{N}, X_n(\omega) = X(\omega) \\ \max\{n \in \mathbb{N}, X_n(\omega) \neq X(\omega)\} & \text{otherwise.} \end{cases}$$ Justify that the application $L$ is well defined.

Let $E$ be an infinite countable subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables defined on $(\Omega, \mathscr{A}, P)$, taking values in $E$, such that for all $\omega \in \Omega$, the sequence $(X_n(\omega))_{n \in \mathbb{N}}$ is stationary and converges to $X(\omega)$. Let $L$ be the random variable defined in 15a. Show that $\mathbf{P}(X_n \neq X) \leqslant \mathbf{P}(L \geqslant n)$ for all integer $n$ in $\mathbb{N}$.

Let $E$ be an infinite countable subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of random variables defined on $(\Omega, \mathscr{A}, P)$, taking values in $E$, such that for all $\omega \in \Omega$, the sequence $(X_n(\omega))_{n \in \mathbb{N}}$ is stationary and converges to $X(\omega)$. Let $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ be the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \; A \in \mathscr{P}(E)\}$. Using the results of 14 and 15b, deduce that $\lim_{n \rightarrow +\infty} \left\|\mu_{X_n} - \mu_X\right\| = 0$.

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 $\sum a_n^2$ converges. Let $S_N = \sum_{n=0}^N X_n a_n$ and let $A_j = \{|S_{\phi(j+1)} - S_{\phi(j)}| > 2^{-j}\}$. Using the sequence $(\phi(j))_{j \in \mathbb{N}}$ satisfying $\sum_{n > \phi(j)}^{+\infty} a_n^2 \leqslant \frac{1}{8^j}$, deduce the bound $\mathbb{P}(A_j) \leqslant 2^{-j}$.

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 $\mathbb{P}(A_j) \leqslant 2^{-j}$.

Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$, and let $S_N = \sum_{n=0}^N X_n a_n$. With the events $$B_{j,m} = \left\{|S_m - S_{\phi(j)}| > 2^{-j} \text{ and } \forall n \in \llbracket \phi(j), m-1 \rrbracket, \; |S_n - S_{\phi(j)}| \leqslant 2^{-j}\right\},$$ $$B_j = \left\{\max_{\phi(j)+1 \leqslant n \leqslant \phi(j+1)} |S_n - S_{\phi(j)}| > 2^{-j}\right\},$$ 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 $$B_j = \bigcup_{\phi(j) < m \leqslant \phi(j+1)} B_{j,m}.$$

To each function $f \in E$, we associate the function $U ( f )$ defined for all $x > 0$ by $U ( f ) ( x ) = \int _ { 0 } ^ { + \infty } \left( \mathrm { e } ^ { \min ( x , t ) } - 1 \right) f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. It has been shown that $| U ( f ) ( x ) | \leqslant 4 \| f \| \frac { \sqrt { x } \mathrm { e } ^ { x / 2 } } { 1 + x }$ for all $x > 0$. Deduce that $$\| U ( f ) \| \leqslant 4 \| f \|.$$

To each function $f \in E$, we associate the function $U ( f )$ defined for all $x > 0$ by $U ( f ) ( x ) = \int _ { 0 } ^ { + \infty } \left( \mathrm { e } ^ { \min ( x , t ) } - 1 \right) f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Show that $U$ is injective.

With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$, events $A_j$, $B_j$, $B_{j,m}$), let $m \in \llbracket \phi(j)+1, \phi(j+1) \rrbracket$. Show that the function $$\left|\begin{array}{lll} \mathbb{R} & \rightarrow & \mathbb{R} \\ \alpha & \mapsto & 2^{\phi(j+1)-\phi(j)} \mathbb{P}\left(\left\{\left|\alpha S_{\phi(j+1)} - \alpha S_m + S_m - S_{\phi(j)}\right| > 2^{-j}\right\} \cap B_{j,m}\right) \end{array}\right.$$ takes values in $\mathbb{N}$ and is even.

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.

Show that there exists a real $a > 0$ such that $$\forall \theta \in [-\pi,\pi], 1-\cos\theta \geq a\theta^2.$$ Deduce that there exist three reals $t_0 > 0$, $\beta > 0$ and $\gamma > 0$ such that, for all $t \in ]0,t_0]$ and all $\theta \in [-\pi,\pi]$, $$\left|\frac{P(e^{-t}e^{i\theta})}{P(e^{-t})}\right| \leq e^{-\beta(t^{-3/2}\theta)^2} \quad \text{or} \quad \left|\frac{P(e^{-t}e^{i\theta})}{P(e^{-t})}\right| \leq e^{-\gamma(t^{-3/2}|\theta|)^{2/3}}.$$

With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$, events $A_j$, $B_j$, $B_{j,m}$), 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 $\alpha = \pm 1$.

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 $\alpha = \pm 1$.