A classic formulation of the Bernstein inequality (from Wikipedia) is as follow:

Let $X_1, \ldots, X_n$ be independent zero-mean random variables. Suppose that $|X_i|\leq M$ almost surely, for all $i$. Then, for all positive $t$,

$\mathbb{P} \left (\sum_{i=1}^n X_i > t \right ) \leq \exp \left ( -\frac{\tfrac{1}{2} t^2}{\sum \mathbb{E} \left[X_j^2 \right ]+\frac{1}{3} Mt} \right ).$

**Question 1:** Can the assumption $|X_i|\leq M$ be transformed into $|X_i|\leq M_i$ where now the bound can be different for each random variable. I would expect then the term $Mt$ in the bound to turn into $\frac{1}{n}\sum_{i=1}^n M_it$. Is there a reference of a paper with such a result? Or is there any reason why such a result does not exist? For instance the Hoeffding bound is always presented with a result depending on the individual bounds $M_i$.

**Question 2:** The Freedman concentration inequality for martingales is a result like the Bernstein bound but where the random variable can be dependent. Can we also have a result for the Freedman inequality with $\frac{1}{n}\sum_i M_i$ appearing instead of $\max_{i} M_i$? Do you know a paper proving that?

**Question 3:** For my research I would actually be happy already if a martingale concentration inequality exists in the special case where the random variable $X_i$ are Bernoulli with parameter $p_i$, $\beta(p_i)$, and then scaled by $\frac{1}{p_i}$, $X_i=\frac{1}{p_i}\beta(p_i)$ (this random variable has a variance and a range of order $\frac{1}{p_i}$ when $p_i$ is small). In my case the value of $p_i$ depend on the previous realizations $X_{i-1},\ldots,X_{i}$.
I would hope the following is true then $\mathbb{P} \left (\sum_{i=1}^n X_i > t \right ) \leq \exp \left ( -\frac{\tfrac{1}{2} t^2}{\sum_i 1/p_i} \right ).$

I am aware of a result by Maurer (see reference below) but there the range $M_i$ appears in the bound as $\sum_{i}M^2_i$.

*Maurer, Andreas*, **A bound on the deviation probability for sums of non-negative random variables**, JIPAM, J. Inequal. Pure Appl. Math. 4, No.1, Paper No.15, 6 p. (2003). ZBL1021.60036.

Thanks for your help!