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Let $(X_{i})_{i\geq 1}$ be a sequence of centered real-valued martingale-differences with respect to some filtration $(\mathcal{F}_{i})_{i \geq 1}$. Define $S_{n} = \sum_{i=1}^{n}X_{i}$ and $\Sigma^{2}=\sum_{i=1}^{n}E[X^2_{i}]$. Furthermore, assume that the variables are a.s. bounded so that $|{X_{i}}| \leq M$ for some positive $M$. Let $h >0$ be some real number.

I am interested in the exponential inequalities of the form

\begin{equation} \label{eq:main_ineq} E[\exp(hS_{n})] \leq \exp(\frac{\Sigma^2}{M^2} (\exp(hM)-hM-1)) \end{equation}

Such upper bound holds for example if $(X_{i})_{i\geq 1}$ are i.i.d. (see ex. Equation 1 in herein book Concentration Inequalities', by Boucheron, Lugosi and Massart page 25-26. ). Also, if we assume natural (for martingale differences) but stronger condition that a.s. holds that $\sum_{i=1}^{n}E_{i-1}[X^2_{i}] \leq \Sigma^{2}$ then the aforementioned inequality also holds (see for example here in the proof of the deviation bound of Theorem 3.2).

But are there results under the condition on the variance (+ maybe some additional assumption which does not involve uniform bound on $E_{i-1}[{X_{i}^2}]$)?

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    $\begingroup$ I think a useful bound not involving assumptions on conditional moments does not exist. However, it is likely a rather challenging and rather thankless task to construct an appropriate counterexample, especially when it is not quite clear what assumptions are allowed here. $\endgroup$ Sep 22, 2019 at 21:50
  • $\begingroup$ @IosifPinelis. Thank you for the comment and sharing your thoughts. Indeed, I formulated my question not clearly. What I was originally interested is the Bennett's (or Bernstein's) type of inequality for deviation of the sum of martingale differences where instead of conditional 2nd moment we have only the bound for variance. $\endgroup$ Sep 24, 2019 at 20:36

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Let $J$ take the value $M$ with probability $1/M^2$ and the value 1 with probability $1-1/M^2$. Let $R_i$ be i.i.d. $\pm 1$ valued random variables of of mean zero, and define $X_i:=JR_i$. Then for $h=1/M$ it is easy to see your inequality is not satisfied, as the LHS is at least $M^{-2}(e/2)^n$, while the RHS is at most $\exp(2n/M^2)$.

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