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 $k \in \{ 1 , \ldots , n \}$, $$P \left[ X _ { k } = 1 \right] = P \left[ X _ { k } = - 1 \right] = \frac { 1 } { 2 }$$ We define $$S _ { n } = \frac { 1 } { n } \sum _ { k = 1 } ^ { n } X _ { k }$$ as well as, for all $\lambda \in \mathbb { R }$, $$\psi ( \lambda ) = \log \left( \frac { 1 } { 2 } e ^ { \lambda } + \frac { 1 } { 2 } e ^ { - \lambda } \right)$$ Show that for all $t \in \mathbb { R }$, we have $$\frac { 1 } { n } \log P \left[ S _ { n } \geqslant t \right] \leqslant \inf _ { \lambda \geqslant 0 } ( \psi ( \lambda ) - \lambda t )$$

Show that $\phi _ { X }$ is continuous on $\mathbb { R }$.

Let $n \geqslant 1$ be a natural integer, and let $(X_1, \ldots, X_n)$ be discrete real random variables that are mutually independent such that, for all $k \in \{1, \ldots, n\}$, $$P[X_k = 1] = P[X_k = -1] = \frac{1}{2}$$ We define $$S_n = \frac{1}{n} \sum_{k=1}^{n} X_k$$ For each $\lambda \geqslant 0$, we set $$m(\lambda) = \frac{E[X_1 \exp(\lambda X_1)]}{E[\exp(\lambda X_1)]}$$ For all $n \geqslant 1$, $\lambda \geqslant 0$ and $\varepsilon > 0$, we denote by $I_n(\lambda, \varepsilon)$ the random variable defined by $$I_n(\lambda, \varepsilon) = \begin{cases} 1 & \text{if } |S_n - m(\lambda)| \leqslant \varepsilon \\ 0 & \text{otherwise.} \end{cases}$$ Show that $$P[|S_n - m(\lambda)| \leqslant \varepsilon] \geqslant E[I_n(\lambda, \varepsilon) \exp(\lambda n(S_n - m(\lambda) - \varepsilon))],$$

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 $k \in \{ 1 , \ldots , n \}$, $$P \left[ X _ { k } = 1 \right] = P \left[ X _ { k } = - 1 \right] = \frac { 1 } { 2 }$$ We define $$S _ { n } = \frac { 1 } { n } \sum _ { k = 1 } ^ { n } X _ { k }$$ For each $\lambda \geqslant 0$, we set $$m ( \lambda ) = \frac { E \left[ X _ { 1 } \exp \left( \lambda X _ { 1 } \right) \right] } { E \left[ \exp \left( \lambda X _ { 1 } \right) \right] }$$ For all $n \geqslant 1 , \lambda \geqslant 0$ and $\varepsilon > 0$, we denote by $I _ { n } ( \lambda , \varepsilon )$ the random variable defined by $$I _ { n } ( \lambda , \varepsilon ) = \begin{cases} 1 & \text { if } \left| S _ { n } - m ( \lambda ) \right| \leqslant \varepsilon \\ 0 & \text { otherwise } \end{cases}$$ Show that $$P \left[ \left| S _ { n } - m ( \lambda ) \right| \leqslant \varepsilon \right] \geqslant E \left[ I _ { n } ( \lambda , \varepsilon ) \exp \left( \lambda n \left( S _ { n } - m ( \lambda ) - \varepsilon \right) \right] , \right.$$

Let $n \geqslant 1$ be a natural integer, and let $(X_1, \ldots, X_n)$ be discrete real random variables that are mutually independent such that, for all $k \in \{1, \ldots, n\}$, $$P[X_k = 1] = P[X_k = -1] = \frac{1}{2}$$ We define $$S_n = \frac{1}{n} \sum_{k=1}^{n} X_k$$ as well as, for all $\lambda \in \mathbb{R}$, $$\psi(\lambda) = \log\left(\frac{1}{2}e^{\lambda} + \frac{1}{2}e^{-\lambda}\right)$$ For each $\lambda \geqslant 0$, we set $$m(\lambda) = \frac{E[X_1 \exp(\lambda X_1)]}{E[\exp(\lambda X_1)]}$$ as well as $$D_n(\lambda) = \exp(\lambda n S_n - n \psi(\lambda))$$ For all $n \geqslant 1$, $\lambda \geqslant 0$ and $\varepsilon > 0$, we denote by $I_n(\lambda, \varepsilon)$ the random variable defined by $$I_n(\lambda, \varepsilon) = \begin{cases} 1 & \text{if } |S_n - m(\lambda)| \leqslant \varepsilon \\ 0 & \text{otherwise.} \end{cases}$$ Show that $$E[I_n(\lambda, \varepsilon) D_n(\lambda)] \geqslant 1 - \frac{4}{n\varepsilon^2}$$

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 $k \in \{ 1 , \ldots , n \}$, $$P \left[ X _ { k } = 1 \right] = P \left[ X _ { k } = - 1 \right] = \frac { 1 } { 2 }$$ We define $$S _ { n } = \frac { 1 } { n } \sum _ { k = 1 } ^ { n } X _ { k }$$ as well as, for all $\lambda \in \mathbb { R }$, $$\psi ( \lambda ) = \log \left( \frac { 1 } { 2 } e ^ { \lambda } + \frac { 1 } { 2 } e ^ { - \lambda } \right)$$ For each $\lambda \geqslant 0$, we set $$m ( \lambda ) = \frac { E \left[ X _ { 1 } \exp \left( \lambda X _ { 1 } \right) \right] } { E \left[ \exp \left( \lambda X _ { 1 } \right) \right] }$$ as well as $$D _ { n } ( \lambda ) = \exp \left( \lambda n S _ { n } - n \psi ( \lambda ) \right)$$ For all $n \geqslant 1 , \lambda \geqslant 0$ and $\varepsilon > 0$, we denote by $I _ { n } ( \lambda , \varepsilon )$ the random variable defined by $$I _ { n } ( \lambda , \varepsilon ) = \begin{cases} 1 & \text { if } \left| S _ { n } - m ( \lambda ) \right| \leqslant \varepsilon \\ 0 & \text { otherwise } \end{cases}$$ Show that $$E \left[ I _ { n } ( \lambda , \varepsilon ) D _ { n } ( \lambda ) \right] \geqslant 1 - \frac { 4 } { n \varepsilon ^ { 2 } }$$

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 $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. Let $k \in \mathbb { N } ^ { * }$. We assume that $X$ admits a moment of order $k$. Let $j$ be an integer such that $1 \leqslant j \leqslant k$. Show that for all real $x , | x | ^ { j } \leqslant 1 + | x | ^ { k }$ and deduce that $X$ admits a moment of order $j$.

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 fix a natural integer $k \in \mathbb { N }$ and assume both that $\phi _ { X }$ is of class $C ^ { 2 k + 2 }$ on $\mathbb { R }$ and that $X$ admits a moment of order $2 k$. We denote $\alpha = \mathbb { E } \left( X ^ { 2 k } \right)$. What can we say about $X$ if $\alpha$ is zero ?

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 fix a natural integer $k \in \mathbb { N }$ and assume both that $\phi _ { X }$ is of class $C ^ { 2 k + 2 }$ on $\mathbb { R }$ and that $X$ admits a moment of order $2 k$. We denote $\alpha = \mathbb { E } \left( X ^ { 2 k } \right)$ and assume that $\alpha > 0$. Let $Y : \Omega \rightarrow \mathbb { R }$ be a random variable satisfying $Y ( \Omega ) = X ( \Omega )$ and, for all $n \in \mathbb { N }$, $$\mathbb { P } \left( Y = x _ { n } \right) = \frac { a _ { n } x _ { n } ^ { 2 k } } { \alpha }$$ Show that $\phi _ { Y }$ is of class $C ^ { 2 }$ on $\mathbb { R }$.

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 fix a natural integer $k \in \mathbb { N }$ and assume both that $\phi _ { X }$ is of class $C ^ { 2 k + 2 }$ on $\mathbb { R }$ and that $X$ admits a moment of order $2 k$. We denote $\alpha = \mathbb { E } \left( X ^ { 2 k } \right)$ and assume that $\alpha > 0$. Using the result of Q35, deduce that $X$ admits a moment of order $2 k + 2$.

We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set. Let $k \in \mathbb { N } ^ { * }$. Deduce from the previous questions that if $\phi _ { X }$ is of class $C ^ { 2 k }$ on $\mathbb { R }$, then $X$ admits a moment of order $2 k$.

What relation exists between $S _ { n }$ and $Y _ { n }$ ? Deduce the expectation and variance of $S _ { n }$. Justify that $S _ { n }$ and $n$ have the same parity.

Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$. If $n \in \mathbb{N}^*$, we set $g(n) = r_1(n) - r_3(n)$ where $r_i(n) = \operatorname{Card}\{d \in \mathbb{N} : d \equiv i [4] \text{ and } d \mid n\}$.
Show that $E(g(X)) = \lim_{n \rightarrow +\infty} \prod_{k=1}^{n} E\left(g\left(p_k^{\nu_{p_k}(X)}\right)\right)$.

Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$.
Show that if $p$ is a prime number such that $p \equiv 1 [4]$, we have $$E\left(g\left(p^{\nu_p(X)}\right)\right) = \frac{1}{1 - p^{-s}}.$$

Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$.
Calculate $E\left(g\left(p^{\nu_p(X)}\right)\right)$ if $p$ is a prime number satisfying $p \equiv 3 [4]$.

Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$.
Deduce $$E(g(X)) = \lim_{n \rightarrow +\infty} \prod_{k=1}^{n} \frac{1}{1 - \chi_4\left(p_k\right) p_k^{-s}}.$$

Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$. We recall that $\chi_4(2n) = 0$ and $\chi_4(2n-1) = (-1)^{n-1}$ for $n \in \mathbb{N}^*$.
Show that, if $p$ is a prime number, $$E\left(\chi_4\left(p^{\nu_p(X)}\right)\right) = \frac{1 - p^{-s}}{1 - \chi_4(p) p^{-s}}.$$

Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$. We recall that $\chi_4(2n) = 0$ and $\chi_4(2n-1) = (-1)^{n-1}$ for $n \in \mathbb{N}^*$.
Show that $$E\left(\chi_4(X)\right) = \frac{1}{\zeta(s)} \lim_{n \rightarrow +\infty} \prod_{k=1}^{n} \frac{1}{1 - \chi_4\left(p_k\right) p_k^{-s}}.$$

Show that if $p = \frac { 1 } { 2 }$, then $T$ does not admit an expectation.

Let $X$ be a discrete random variable with finite expectation. Show that $$\mathbb{E}\left(X \mathbb{1}_{|X| \leqslant C}\right) \xrightarrow{C \rightarrow +\infty} \mathbb{E}(X).$$

Let $X$ be a discrete random variable with finite expectation. Show that $$\mathbb{E}\left(X \mathbb{1}_{|X| \leqslant C}\right) \xrightarrow{C \rightarrow +\infty} \mathbb{E}(X).$$

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 $$\lim_{C \rightarrow +\infty} \sigma_{ij}(C) = 1$$

For all $(i,j) \in (\mathbb{N}^{\star})^{2}$ and for all $C > 0$, we set $\sigma_{ij}(C) = \sqrt{\mathbb{V}\left(X_{ij} \mathbb{1}_{|X_{ij}| \leqslant C}\right)}$.
Deduce that $$\lim_{C \rightarrow +\infty} \sigma_{ij}(C) = 1$$

We consider an urn containing $A$ balls of which $pA$ are white and $qA$ are black. We draw simultaneously $n$ balls from the urn. We number from 1 to $pA$ each of the white balls and, for any natural integer $i \in \llbracket 1, pA \rrbracket$, we define $$Y_i = \begin{cases} 1 & \text{if the ball numbered } i \text{ was drawn,} \\ 0 & \text{otherwise.} \end{cases}$$ Let $Y = \sum_{i=1}^{pA} Y_i$ be the number of white balls drawn. Deduce the value of the variance of $Y$. Compare it to that of $Z$ (where $Z \sim \mathcal{B}(n,p)$).

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 $1 \leqslant i \leqslant j$.