I'm currently trying to figure out the following inequality. It looks like an inequality for the exponential sum, but I can't verify it or find a source explaining it any further. Most likely it has to do with the remainder I guess... $$|E[\exp(itX_{n,k})|F_{n,k-1}]-1-\frac{1}{2}t^2E[X_{n,k}^2|F_{n,k-1}]|\\ \leq \frac{1}{6}|t|^3E[|X_{n,k}|^3\mathrm{1}_{|X_{n,k}|\leq \epsilon}\big{|}F_{n,k-1}]+t^2E[X_{n,k}^21_{X_{n,k}>\epsilon}|F_{n,k-1}]$$
Where $E[X_{n,k}|F_{n,k-1}]=0$ for all $k,n \in \mathbb{N}$
Especially this is taken from Dvoretzky, 1972, ASYMPTOTIC NORMALITY FOR SUMS OF DEPENDENT RANDOM VARIABLES and can be found in the proof of theorem 2.1 equality (4.4). Thanks in advance.
Somebody on MO just pointed out to me that the inequality, especially the first term in the second line comes from the remainder in integral form of the corresponding Taylor series.