Let $X_1, X_2, \dots, X_n$ be a martingale difference sequence such that $$ X_i \leq y \quad \text{and} \quad \sum_{i=1}^{n} Var(X_i) \leq B^2. $$ Question 1: Does the following hold? $$ \mathbb{P}\left[ \sum_{i=1}^{n}X_i \geq x \right] \leq \exp{\left(\frac{-x^2}{2B^2 + \frac{2}{3}xy}\right)}. $$

A similar bound (albeit for independent random variables) is given in Corollary 2 in https://epubs.siam.org/doi/abs/10.1137/1134032. I have seen that exponential inequalities for sums of independent random variables can be extended to martingales generally.

Question 2: If the bound given in question 1 doesn't hold, does any other similar exponential inequality exist for the LHS? I have came across Freedman's inequality (Theorem 1.6 in Freedman (1975)) which deals with similar quantities but it contains $Var(X_i | \mathcal{F}_{i-1})$. As seen from the above, I would rather have the bound in terms of $Var(X_i)$.

Thank you for your time and consideration.

Let $X_1, X_2, \dots, X_n$ be a martingale difference sequence such that $$ X_i \leq y \quad \text{and} \quad \sum_{i=1}^{n} \operatorname{Var}(X_i) \leq B^2. $$ Question 1: Does the following hold? $$ \mathbb{P}\left[ \sum_{i=1}^{n}X_i \geq x \right] \leq \exp{\left(\frac{-x^2}{2B^2 + \frac{2}{3}xy}\right)}. $$

A similar bound (albeit for independent random variables) is given in Corollary 2 in Pinelis–Utev (1990) (DOI link). I have seen that exponential inequalities for sums of independent random variables can be extended to martingales generally.

Question 2: If the bound given in question 1 doesn't hold, does any other similar exponential inequality exist for the LHS? I have came across Freedman's inequality (Theorem 1.6 in Freedman (1975)) which deals with similar quantities but it contains $\operatorname{Var}(X_i | \mathcal{F}_{i-1})$. As seen from the above, I would rather have the bound in terms of $\operatorname{Var}(X_i)$.

Thank you for your time and consideration.

Let $X_1, X_2, \dots, X_n$ be a martingale difference sequence such that $$ X_i \leq y \quad \text{and} \quad \sum_{i=1}^{n} Var(X_i) \leq B^2. $$ Question 1: Does the following hold? $$ \mathbb{P}\left[ \sum_{i=1}^{n}X_i \geq x \right] \leq \exp{\left(\frac{-x^2}{2B^2}\right)}. $$

A similar bound (albeit for independent random variables) is given in Theorem 7 in https://epubs.siam.org/doi/abs/10.1137/1134032. I have seen that exponential inequalities for sums of independent random variables can be extended to martingales generally.

Question 2: If the bound given in question 1 doesn't hold, does any other similar exponential inequality exist for the LHS? I have came across Freedman's inequality (Theorem 1.6 in Freedman (1975)) which deals with similar quantities but it contains $Var(X_i | \mathcal{F}_{i-1})$. As seen from the above, I would rather have the bound in terms of $Var(X_i)$.

Thank you for your time and consideration.

Let $X_1, X_2, \dots, X_n$ be a martingale difference sequence such that $$ X_i \leq y \quad \text{and} \quad \sum_{i=1}^{n} Var(X_i) \leq B^2. $$ Question 1: Does the following hold? $$ \mathbb{P}\left[ \sum_{i=1}^{n}X_i \geq x \right] \leq \exp{\left(\frac{-x^2}{2B^2 + \frac{2}{3}xy}\right)}. $$

A similar bound (albeit for independent random variables) is given in Corollary 2 in https://epubs.siam.org/doi/abs/10.1137/1134032. I have seen that exponential inequalities for sums of independent random variables can be extended to martingales generally.

Question 2: If the bound given in question 1 doesn't hold, does any other similar exponential inequality exist for the LHS? I have came across Freedman's inequality (Theorem 1.6 in Freedman (1975)) which deals with similar quantities but it contains $Var(X_i | \mathcal{F}_{i-1})$. As seen from the above, I would rather have the bound in terms of $Var(X_i)$.

Thank you for your time and consideration.