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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.

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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.

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  • $\begingroup$ 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 $\endgroup$ – DrShredz Jul 24 '18 at 22:33
  • $\begingroup$ 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. $\endgroup$ – Iosif Pinelis Jul 25 '18 at 0:03
  • $\begingroup$ Oh thanks so much! Your answer probably saved Me from going insane over this ... :D $\endgroup$ – DrShredz Jul 25 '18 at 7:53

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