Using the result of Q29, propose another proof of the result obtained in question 6, i.e., the pointwise limit of the sequence of functions $(\varphi_n)_{n \geqslant 1}$ defined by $$\forall n \in \mathbb{N}^{\star}, \quad \varphi_n(t) = \mathbb{E}(\cos(t X_n)).$$

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, $(U_n)_{n \geqslant 1}$ a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $$\forall n \in \mathbb{N}^{\star}, \quad Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}.$$ For every continuous function $f$ from $[0,1]$ to $\mathbb{R}$, the sequence $(\mathbb{E}(f(Y_n)))_{n \geqslant 1}$ converges to $\int_0^1 f(t)\,\mathrm{d}t$.
Using the previous result, propose another proof of the result obtained in question 6.

Let $(X_1, \ldots, X_n)$ be discrete real random variables that are mutually independent such that, for all $k \in \{1, \ldots, n\}$, $$P[X_k = 1] = P[X_k = -1] = \frac{1}{2}$$ For each $\lambda \geqslant 0$, we set $$m(\lambda) = \frac{E[X_1 \exp(\lambda X_1)]}{E[\exp(\lambda X_1)]}$$ Show that the function $m$ is strictly increasing on $\mathbb{R}_{\geqslant 0}$, and that for all $t \in [0,1]$, there exists a unique $\lambda \geqslant 0$ such that $m(\lambda) = t$.

Let $n \geqslant 1$ be a natural integer, and let $\left( X _ { 1 } , \ldots , X _ { n } \right)$ be mutually independent discrete real random variables such that, for all $k \in \{ 1 , \ldots , n \}$, $$P \left[ X _ { k } = 1 \right] = P \left[ X _ { k } = - 1 \right] = \frac { 1 } { 2 }$$ For each $\lambda \geqslant 0$, we set $$m ( \lambda ) = \frac { E \left[ X _ { 1 } \exp \left( \lambda X _ { 1 } \right) \right] } { E \left[ \exp \left( \lambda X _ { 1 } \right) \right] }$$ Show that the function $m$ is strictly increasing on $\mathbb { R } _ { + }$, and that for all $t \in [ 0,1 [$, there exists a unique $\lambda \geqslant 0$ such that $m ( \lambda ) = t$.

Let $a$ and $b$ be two real numbers and $Y = a X + b$. For all real $t$, express $\phi _ { Y } ( t )$ in terms of $\phi _ { X } , t , a$ and $b$.

Let $t \in \mathbb { R }$. Give an expression of $\phi _ { X } ( - t )$ in terms of $\phi _ { X } ( t )$. Deduce a necessary and sufficient condition on the image $\phi _ { X } ( \mathbb { R } )$ for the function $\phi _ { X }$ to be even.

We assume that there exists $t _ { 0 } \in \mathbb { R } ^ { * }$ such that $\left| \phi _ { X } \left( t _ { 0 } \right) \right| = 1$. We assume further that $X ( \Omega )$ is countable and we use the notations of question 2 (i.e. $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ with $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$). Deduce that $\mathbb { P } \left( X \in a + \frac { 2 \pi } { t _ { 0 } } \mathbb { Z } \right) = 1$.

Let $X$ be a real and discrete random variable and $m \in \mathbb { R }$. For $T \in \mathbb { R } _ { + } ^ { * }$, we set $V _ { m } ( T ) = \frac { 1 } { 2 T } \int _ { - T } ^ { T } \phi _ { X } ( t ) \mathrm { e } ^ { - \mathrm { i } m t } \mathrm {~d} t$. We assume that $X ( \Omega )$ is finite and we use the notations of question 1: $X ( \Omega ) = \left\{ x _ { 1 } , \ldots , x _ { r } \right\}$ with $a _ { k } = \mathbb { P } \left( X = x _ { k } \right)$. Show that, for all $T \in \mathbb { R } _ { + } ^ { * }$, we have $V _ { m } ( T ) = \sum _ { n = 1 } ^ { r } \operatorname { sinc } \left( T \left( x _ { n } - m \right) \right) \mathbb { P } \left( X = x _ { n } \right)$.

Let $X$ be a real and discrete random variable and $m \in \mathbb { R }$. For $T \in \mathbb { R } _ { + } ^ { * }$, we set $V _ { m } ( T ) = \frac { 1 } { 2 T } \int _ { - T } ^ { T } \phi _ { X } ( t ) \mathrm { e } ^ { - \mathrm { i } m t } \mathrm {~d} t$. We assume that $X ( \Omega )$ is finite and we use the notations of question 1: $X ( \Omega ) = \left\{ x _ { 1 } , \ldots , x _ { r } \right\}$ with $a _ { k } = \mathbb { P } \left( X = x _ { k } \right)$. Using the result of Q15, deduce that $V _ { m } ( T ) \xrightarrow [ T \rightarrow + \infty ] { } \mathbb { P } ( X = m )$.

Let $X$ be a real and discrete random variable and $m \in \mathbb { R }$. For $T \in \mathbb { R } _ { + } ^ { * }$, we set $V _ { m } ( T ) = \frac { 1 } { 2 T } \int _ { - T } ^ { T } \phi _ { X } ( t ) \mathrm { e } ^ { - \mathrm { i } m t } \mathrm {~d} t$. We assume that $X ( \Omega )$ is countable and we use the notations of question 2: $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ with $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. For $n \in \mathbb { N }$ and $h \in \mathbb { R } _ { + } ^ { * }$, we set $g _ { n } ( h ) = \operatorname { sinc } \left( \frac { x _ { n } - m } { h } \right) \mathbb { P } \left( X = x _ { n } \right)$. Show that for all $T \in \mathbb { R } _ { + } ^ { * }$, we have $V _ { m } ( T ) = \sum _ { n = 0 } ^ { + \infty } g _ { n } \left( \frac { 1 } { T } \right)$.

We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set, with $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ and $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. We assume that $\phi _ { X }$ is of class $C ^ { 2 }$ on $\mathbb { R }$. We denote $f$ the function which to all real $h > 0$ associates $f ( h ) = \frac { 2 \phi _ { X } ( 0 ) - \phi _ { X } ( 2 h ) - \phi _ { X } ( - 2 h ) } { 4 h ^ { 2 } }$. What is the limit of $f$ at 0 ?

We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set, with $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ and $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. We assume that $\phi _ { X }$ is of class $C ^ { 2 }$ on $\mathbb { R }$. We denote $f ( h ) = \frac { 2 \phi _ { X } ( 0 ) - \phi _ { X } ( 2 h ) - \phi _ { X } ( - 2 h ) } { 4 h ^ { 2 } }$ for $h > 0$. Show that for all $h \in \mathbb { R } ^ { * } , f ( h ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } \frac { \sin ^ { 2 } \left( h x _ { n } \right) } { h ^ { 2 } }$.

Let $X _ { 1 } , \ldots , X _ { n } , Y _ { 1 } , \ldots , Y _ { n }$ be mutually independent random variables with the same distribution $\mathcal { R }$ (where $X(\Omega) = \{-1,1\}$, $\mathbb{P}(X=-1)=\mathbb{P}(X=1)=\frac{1}{2}$). We define the random vectors $X = \frac { 1 } { \sqrt { n } } \left( X _ { 1 } , \ldots , X _ { n } \right) ^ { \top }$ and $Y = \frac { 1 } { \sqrt { n } } \left( Y _ { 1 } , \ldots , Y _ { n } \right) ^ { \top }$ taking values in $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$.
Prove that, for every real number $t$, $$\mathbb { E } ( \exp ( t \langle X \mid Y \rangle ) ) = \left( \operatorname { ch } \left( \frac { t } { n } \right) \right) ^ { n }$$