Exponential upper bounds for sums of martingale differences

Question

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 $E_{i-1}[{X_{i}^2}]$)?

Answer 1

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 $\exp(2n/M^2)$.