# 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.

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 $$|\cos t-1-t^2/2|\le\tfrac16\,|t|^3,$$ 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.
• Thanks so much for the detailed answer, but I'm still struggling why it holds that $|E_F \exp(itX)-1-\frac{1}{2}t^2E_FX^2|=|E_F r(itX)|$. This isn't simply inserting since it would give me $exp(-tX)$ instead of $exp(itX)$ but inserting $tx$ doesn't give me the correct result either since I then get $+\frac{1}{2}t^2E_FX^2$. This is why I'm confused that the term to be estimated is some Taylor-remainder as the signs never match. At this point I'm feeling as if I somehow missed something trivial on complex numbers… :D – DrShredz Jul 24 '18 at 22:33
• In fact, there is a typo in the Dvoretzky paper, as is explained now in my answer. Also, there was a typo in my answer as well: I had to write $r(tX)$ instead of $r(itX)$; this is now corrected as well. – Iosif Pinelis Jul 25 '18 at 0:03