Show that, if there exist $a \in \mathbb { R }$ and $t _ { 0 } \in \mathbb { R } ^ { * }$ such that $X ( \Omega ) \subset a + \frac { 2 \pi } { t _ { 0 } } \mathbb { Z }$, then $\left| \phi _ { X } \left( t _ { 0 } \right) \right| = 1$.

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 $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$). Show that there exists $a \in \mathbb { R }$ such that $\sum _ { n = 0 } ^ { + \infty } a _ { n } \exp \left( \mathrm { i } \left( t _ { 0 } x _ { n } - t _ { 0 } a \right) \right) = 1$.

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 $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$). Using the result of Q11, deduce that $\sum _ { n = 0 } ^ { + \infty } a _ { n } \left( 1 - \cos \left( t _ { 0 } x _ { n } - t _ { 0 } a \right) \right) = 0$.

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 $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$). Show that for all $n \in \mathbb { N }$, if $a _ { n } \neq 0$, then $x _ { n } \in a + \frac { 2 \pi } { t _ { 0 } } \mathbb { Z }$.

We admit that $\int _ { 0 } ^ { + \infty } \operatorname { sinc } ( s ) \mathrm { d } s = \frac { \pi } { 2 }$. If $a$ and $b$ are two real numbers, we denote $K _ { a , b }$ the function defined for all real $t$ by $K _ { a , b } ( t ) = \begin{cases} \frac { \mathrm { e } ^ { \mathrm { i } t b } - \mathrm { e } ^ { \mathrm { i } t a } } { 2 \mathrm { i } t } & \text { if } t \neq 0 , \\ \frac { b - a } { 2 } & \text { if } t = 0 . \end{cases}$ Let $X : \Omega \rightarrow \mathbb { R }$ be a random variable such that $X ( \Omega )$ is finite. We assume that the real numbers $a$ and $b$ do not belong to $X ( \Omega )$. Show that $$\frac { 1 } { \pi } \int _ { - N } ^ { N } \phi _ { X } ( - t ) K _ { a , b } ( t ) \mathrm { d } t \xrightarrow [ N \rightarrow + \infty ] { } \mathbb { P } ( a < X < b )$$