Pointwise limit of MGFs or characteristic functions (convergence in distribution)
The question asks to compute the pointwise limit of a sequence of MGFs or characteristic functions, typically to establish convergence in distribution (e.g., Poisson approximation, CLT).
For $n \in \mathbb { N } ^ { * }$, $U _ { n }$ is a random variable on $(\Omega , \mathcal { A } , \mathbb { P })$ following the uniform distribution on $\llbracket 1 , n \rrbracket$. We set $Y _ { n } = \frac { 1 } { n } U _ { n }$. For $t \in \mathbb { R }$, calculate $\lim _ { n \rightarrow + \infty } M _ { Y _ { n } } ( t )$.