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