I need to show that the CF of a two-sided Pareto distribution,
whose pdf is given by $f(x) = |x|^{-3} \mathbb{1}_{|x| \geq 1}$,
is $\phi_X(t) = 1 - t^2(\ln(\frac{1}{|t|}+O(1))$ when $t$ is close to $0$.
I used polynomial expansion of $e^{itX}$ to get:
$E(e^{itX}) = E(1 + itX + \frac{(-1)t^2X^2}{2} + o(t^2)) = 1-\frac{t^2}{2}E(X^2)+o(t^2)$
I am not quite familiar with the big O small o notation, so please correct me if I used them incorrectly.
So I need to get an estimate of $E(X^2)$, and I tried a truncated $X$, $X\mathbb{1}_{X<1/|t|}$.
With this truncated $X$, I am able to calculate its second moment, which is $2\ln\frac{1}{|t|}$
I am basically done, but I am stuck at the last step. In order to finish, I need to get an estimate of $E(X^2\mathbb{1}_{X>1/|t|})$
Can I get some suggestions on how I can proceed? Thanks!