Consider a probability density $f(X)$ of some random variable $X\leq0$, with moments $\mu_n'=\left< X ^n \right>$ and cumulants $\kappa_n$.
I aim to proof
$$ \Delta_n := (-1)^n \left(\mu_n'- \frac{\kappa_n}{(n-2)!} \right) \geq 0 \quad \text{for }n=(2,3,\ldots,\infty) $$
which I have been able to confirm for the first two coefficients,
$$ \Delta_2 = \mu_2' - \kappa_2 = \mu_2' -(\mu_2' - \mu_1'^2) = \mu_1'^2 \geq 0$$
$$ \Delta_3 = -(\mu_3' - \kappa_3) = -(\mu_3' - (\mu_3' - 3 \mu_2'\mu_1' + 2\mu_1')) =\mu_1'(2\mu_1'^2 - 3\mu_2') \geq 0 \quad (\text{since }\mu_2' \geq \mu_1'^2) $$
where I introduced the central moments $\mu_n=\left< (X -\left<X\right>)^n \right>$.
Any help would be very much appreciated.
Note that $(-1)^n(\mu_n' - \kappa_n) \geq 0$ would also proof the above, though being a somewhat stricter requirement.
