Characteristic function say something about the expectation and variance

Question

Show that if $\lim_{t \downarrow 0} (\varphi(t) -1) / t^2 = c > -\infty$ then $EX = 0$ and $E|X|^2 = -2c < \infty$. In particular, if $\varphi(t) = 1 + o(t^2)$, then $\varphi(t) \equiv 1$. Where $\varphi(t)$ is the characteristic function of $X$.

Any idea is appreciated.

Answer 1

Hint: Use theorem 3.3.9 followed by 3.3.8 in Durrett's text.

(I assume you are working from this text, as your last two questions are directly from it)