Let \(X\) be a random variable with characteristic function \(\psi\). Suppose that \(E\left(|X|^{n}ight)<\infty\) for some \(n \in\{1,2, \ldots\}\) and that \(\psi^{(k)}\) exists and is uniformly continuous for \(k \in\{1,2, \ldots, n\}\).

a. Prove that

\[\psi(t)=1+\sum_{k=1}^{n} \frac{\mu_{k}^{\prime}(i t)^{k}}{k !}+o\left(|t|^{n}ight)\]

as \(t ightarrow 0\).

b. Prove that

\[\left.\frac{d^{k} \psi(t)}{d t^{k}}ight|_{t=0}=i^{k} \mu_{k}^{\prime}\]