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 $k$. Deduce an expression of $\mathbb { E } \left( X ^ { k } \right)$ in terms of $\phi _ { X } ^ { ( k ) } ( 0 )$.

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 $f$ 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 $h \in \mathbb { R } ^ { * } , f ( h ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } \frac { \sin ^ { 2 } \left( h x _ { n } \right) } { h ^ { 2 } }$.

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 $\mathbb { R }$. Using the results of Q31 and Q32, deduce that $X$ admits a moment of order 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 $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. 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 $2 k$.

We set, for all $t \in I$, $$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$ The generating series of $T$ is given by $G _ { T } ( t ) = \sum _ { n = 0 } ^ { \infty } \mathbb { P } ( T = n ) t ^ { n }$, defined if $t \in [ - 1,1 ]$. Using the previous questions, express $G _ { T }$ using $g$ and $\mathbb { P } ( T = 0 )$.

We set, for all $t \in I$, $$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$ Deduce that if $p \neq \frac { 1 } { 2 }$, then $T$ admits an expectation.

We set, for all $t \in I$, $$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$ Deduce that $\mathbb { P } ( T \neq 0 ) = 1 - \sqrt { 1 - 4 p ( 1 - p ) }$. Interpret this result when $p = \frac { 1 } { 2 }$.

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

From the previous question, recover the result of questions 11 and 16.

Let $X$ be a real random variable. Show that $\left| \Phi _ { X } ( \theta ) \right| \leq 1$ for all real $\theta$.

In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$.
Show that for all $( a , b ) \in \mathbf { R } ^ { 2 }$ and all real $\theta$,
$$\Phi _ { a X + b } ( \theta ) = \frac { p e ^ { i ( a + b ) \theta } } { 1 - q e ^ { i a \theta } }$$

In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$.
Show that for all $k \in \mathbf { N }$, the random variable $X ^ { k }$ has finite expectation. Show that $\Phi _ { X }$ is of class $\mathcal { C } ^ { \infty }$ on $\mathbf { R }$ and that $\Phi _ { X } ^ { ( k ) } ( 0 ) = i ^ { k } \mathbf { E } \left( X ^ { k } \right)$ for all $k \in \mathbf { N }$.

In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$.
Show that there exists a sequence $\left( P _ { k } \right) _ { k \in \mathbf { N } }$ of polynomials with coefficients in $\mathbf { C }$, independent of $p$, such that
$$\forall \theta \in \mathbf { R } , \forall k \in \mathbf { N } , \Phi _ { X } ^ { ( k ) } ( \theta ) = p i ^ { k } e ^ { i \theta } \frac { P _ { k } \left( q e ^ { i \theta } \right) } { \left( 1 - q e ^ { i \theta } \right) ^ { k + 1 } } \quad \text { and } \quad P _ { k } ( 0 ) = 1$$

In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$.
Deduce that there exists a sequence $\left( C _ { k } \right) _ { k \in \mathbf { N } }$ of strictly positive reals, independent of $p$, such that
$$\forall k \in \mathbf { N } , \left| \mathbf { E } \left( X ^ { k } \right) - \frac { 1 } { p ^ { k } } \right| \leq \frac { C _ { k } q } { p ^ { k } }$$

We denote by $G_{X_n}$ and $G_Y$ the generating functions of the variables $X_n$ and $Y$ from the previous question, respectively. Express $G_{X_n}(s)$ as a sum, for $s$ real, and verify that $$\forall s \in \mathbf{R} \quad \lim_{n \rightarrow +\infty} G_{X_n}(s) = G_Y(s)$$