Can Hoeffding's inequality be applied to independent random variables or must they be IID?

Question

I want to clarify the assumptions for random variables for which we can apply Hoeffding's inequality. I know that they have to be independent and both upper- and lower-bounded. However, I've seen proofs and lectures that say they have to be independent AND identically distributed (bottom of first page here, first theorem here), while others just require independence (theorem 4 on pg. 6 here, first page here, section 3 on pg. 2 here). Which is correct? Thanks so much in advance!

Answer 1

There are many variants of it. Consider random variables $X_1$, $X_2$, $\dots$ . Also, assume that for all $i \in \mathbb{N}$, $|X_i|\leq b$ with probability one and $\mathbb{E}[X_i]=0$. If $X_i$s are independent, then, for every $\delta$ and $n$, with probability at least $1-\delta$ we have \begin{align} \frac{1}{n} \sum_{i=1}^{n} X_i \leq b \sqrt{\frac{2 \log(2/\delta)}{n}}. \end{align}

To obtain a bound like this, even independence is not required. For instance, you can take a look at Azuma's inequality. Also, the boundedness can be relaxed using the notion of Subgaussian tails (here).