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.



Inequality for exponential sum in Dvoretzky 1972
First here, there is a typo in the Dvoretzky paper: there must be $-1+\frac{1}{2}t^2E[X\_{n,k}^2|F\_{n,k-1}]$ instead of $-1-\frac{1}{2}t^2E[X\_{n,k}^2|F\_{n,k-1}]$ there. Otherwise, the inequality will not be true in general. Indeed, let, for brevity, $X:=X\_{n,k}$, $F:=F\_{n,k-1}$, $E\_F Z:=E(Z|F)$, and $c:=\epsilon$. Suppose, e.g., that $X$ is independent of $F$, $P(X=1)=P(X=-1)=1/2$, and $c=1$. Then the erroneous inequality becomes 
\begin{equation}
 |\cos t-1-t^2/2|\le\tfrac16\,|t|^3, 
\end{equation}
which is false for small enough $|t|$, since $|\cos t-1-t^2/2|\sim t^2$ as $t\to0$. 


So, we actually need to show that 
$$|E\_F\exp(itX)-1+\tfrac12\,t^2E\_F X^2|\le \tfrac16\,|t|^3 E\_F|X|^3\,1\_{|X|\le c}+t^2E\_FX^2 1\_{|X|>c} $$
given that $E\_F X=0$. 


By [Taylor's theorem with the integral form of the remainder](https://en.wikipedia.org/wiki/Taylor%27s_theorem), 
\begin{equation\*}
 |e^{ix}-1-ix-(ix)^2/2|\le|x|^3/6 \tag{1}
\end{equation\*}
and 
\begin{equation\*}
 |e^{ix}-1-ix|\le x^2/2 
\end{equation\*} 
for real $x$. 
The latter inequality also implies 
\begin{equation\*}
 |e^{ix}-1-ix-(ix)^2/2|\le|e^{ix}-1-ix|+|(ix)^2/2|\le x^2/2+x^2/2=x^2. \tag{2}
\end{equation\*}


Let $r(x):=e^{ix}-1-ix-(ix)^2/2$ and write, in view of (1) and (2):
\begin{multline\*}
 |E\_F\exp(itX)-1+\tfrac12\,t^2E\_F X^2|
 =|E\_F r(tX)|\le|E\_F r(tX)\,1\_{|X|\le c}|+|E\_F r(tX)\,1\_{|X|>c}| \\ 
 \le\tfrac16\,|t|^3 E\_F|X|^3\,1\_{|X|\le c}+t^2E\_FX^2 1\_{|X|>c},
\end{multline\*}
as desired. 


