Bernstein’s Inequality can be stated as follows : Let $x_1, x_2, \dots, x_n$ be independent bounded random variables such that $\mathbb{E}[x_i] = 0$ and $|x_i| \leq \zeta$ with probability $1$ and let $\sigma^2 = \tfrac{1}{n}\sum_{1}^{n} Var\{x_i\}$. Then for any $\epsilon > 0$, we have $$ \mathbb{P} \left[ \frac{1}{n} \sum_{i=1}^{n} x_i \geq \epsilon \right] \leq \exp{\left\{ \frac{-n \epsilon^2}{ 2 \sigma^2 + 2\zeta \epsilon/3} \right\}} $$

If instead of $|x_i| \leq \zeta$ with probability $1$, it is the case that $ \sum_{i=1}^{n} \mathbb{P}\left\{ |x_i| > \zeta \right\} \leq \delta$, then is the following applicable?

$$ \mathbb{P} \left[ \frac{1}{n} \sum_{i=1}^{n} x_i \geq \epsilon \right] \leq \exp{\left\{ \frac{-n \epsilon^2}{ 2 \sigma^2 + 2\zeta \epsilon/3} \right\}} + \delta \qquad \qquad \qquad \qquad (1) $$

I think, the above extension is similar to the extension of the Azuma-Hoeffding inequality proved in Theorem 32 of Chung and Lu(2006). The question concerning the extension of the Azuma-Hoeffding inequality was also asked here.

Is it possible to extend Bernstein’s Inequality as eq. $(1)$ following the lines of the proof of Theorem 32 in Chung and Lu(2006)?

Addendum : The Freedman inequality for martingales is a result like the Bernstein inequality but where the random variables can be dependent. Is an extension like the above possible for Freedman inequality?

For reference Freedman inequality (Theorem 1.6 in Freedman (1975)] ) can be stated as follows: let $x_1, x_2, \dots $ be a martingale difference sequence and $|x_i| \leq \zeta$ for all $i$. Then $$ \mathbb{P} \left[ \sum_{i=1}^{n} x_i \geq \epsilon, \sum_{i=1}^{n} var(x_i | \mathcal{F}_{i-1}) \leq \eta \right] \leq \exp{\left\{ \frac{-\epsilon^2}{2\eta + 2 \zeta \epsilon/3 } \right\}}. $$

If instead of $|x_i| \leq \zeta$ with probability $1$, it is the case that $ \sum_{i=1}^{n} \mathbb{P}\left\{ |x_i| > \zeta \right\} \leq \delta$, then is the following applicable? $$ \mathbb{P} \left[ \sum_{i=1}^{n} x_i \geq \epsilon, \sum_{i=1}^{n} var(x_i | \mathcal{F}_{i-1}) \leq \eta \right] \leq \exp{\left\{ \frac{-\epsilon^2}{2\eta + 2 \zeta \epsilon/3 } \right\}} + \delta \qquad \qquad (2) $$ Thank you.

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