Inequality for exponential sum in Dvoretzky 1972 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.
 A: 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, 
\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. 
