Given a real $t > 0$, we set, following the notations of part $\mathbf{C}$,
$$m _ { t } : = S _ { 1,1 } ( t ) \quad \text { and } \quad \sigma _ { t } : = \sqrt { S _ { 2,1 } ( t ) }$$
Given reals $t > 0$ and $\theta$, we set
$$h ( t , \theta ) = e ^ { - i m _ { t } \theta } \frac { P \left( e ^ { - t } e ^ { i \theta } \right) } { P \left( e ^ { - t } \right) }$$
Let $\theta \in \mathbf { R }$ and $t \in \mathbf { R } _ { + } ^ { * }$. We consider, for all $k \in \mathbf { N } ^ { * }$, a random variable $Z _ { k }$ following the distribution $\mathcal { G } \left( 1 - e ^ { - k t } \right)$, and we set $Y _ { k } = k \left( Z _ { k } - \mathrm { E } \left( Z _ { k } \right) \right)$. Prove that
$$h ( t , \theta ) = \lim _ { n \rightarrow + \infty } \prod _ { k = 1 } ^ { n } \Phi _ { Y _ { k } } ( \theta )$$
Deduce, using in particular question $21 \triangleright$, the inequality
$$\left| h ( t , \theta ) - e ^ { - \frac { \sigma _ { t } ^ { 2 } \theta ^ { 2 } } { 2 } } \right| \leq K ^ { 3 / 4 } | \theta | ^ { 3 } S _ { 3,3 / 4 } ( t ) + K \theta ^ { 4 } S _ { 4,1 } ( t )$$