Let $X$ be a random variable on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and $\varphi_{X}$ the corresponding characteristic function.
I try to show: \begin{equation} \mathbb{E}(X^2)=\lim_{t \to 0} \frac{1}{t^2}(2-\varphi_{X}(t)-\varphi_{X}(-t)), \end{equation} where the second moment doesn't have to be finite.
I am grateful for any help. Thanks in advance!
$2-\phi_X(t)-\phi_X(-t)=E2(1-\cos (tX))$. Using the inequality $1-\cos (t) \leq \frac {t^{2}} 2$ we get $2-\phi_X(t)-\phi_X(-t) \leq t^{2}EX^{2}$ which gives $\lim \frac {2-\phi_X(t)-\phi_X(-t)} {t^{2}} \leq EX^{2}$. The reverse inequality follows by Fatou' Lemma: $EX^{2} =2E\lim\inf \frac {1-\cos (tX)} {t^{2}} \leq \lim \inf 2E \frac {1-\cos (tX)} {t^{2}}=\lim \inf \frac {2-\phi_X(t)-\phi_X(-t)} {t^{2}} $.