If the characteristics function of a random variable is differentiable even times then it has finite moment of even order

Question

If the characteristics function of a random variable is differentiable $2n$ times then it has finite moment up to even order $2n$. We know the converse is correct, but how can we prove this statement?

Answer 1

It suffices to consider the case $n=1$ (for $n \geq 2$ use induction). Denote by $\phi$ the characteristic function and $\mu$ the distribution of the random variable $X$. Then

$$\begin{align*} \phi(2h)-2\phi(0)+\phi(-2h) &= \int (e^{-\imath \, 2h x} -2 + e^{\imath \, 2hx}) \, \mu(dx) \\ &= 2 \int (\cos(2hx)-1) \mu(dx). \tag{1} \end{align*}$$ Since \begin{equation*} \frac{1-\cos(2y)}{4y^2} \xrightarrow[]{y \to 0} \frac{1}{2} \end{equation*} we obtain by applying Fatou's lemma \begin{align*} \int x^2 \frac{1}{2} \, d\mu(x) &= \int x^2 \lim_{h \to 0} \frac{1-\cos(2hx)}{4(hx)^2} \, \mu(dx) \\ &\leq \liminf_{h \to 0} \frac{1}{4h^2} \int (1-\cos(2hx)) \, \mu(dx) \\ &\stackrel{(1)}{=} - \frac{1}{2} \liminf_{h \to 0} \frac{1}{4 h^2} \big(\phi(2h)-2\phi(0)+\phi(-2h)) \\ &= - \frac{1}{2} \phi''(0)<\infty. \end{align*}