I am looking for an extension of Azuma's inequality which involves the expected sum of squared differences. In particular, recall that Azuma's inequality states \begin{align*} \Pr[X_n-X_0 \geq a] \leq \exp\big(\frac{-a^2}{\sum_{k=1}^n c_k^2}\big) \end{align*} I am looking for a similar version which involves $\mathbb{E} \sum_{i=1}^n c_i^2$, while we can still enjoy the extra assumption of $0\leq c_i\leq 1$ if it is needed. Any comments/ideas are appreciated.
2026-03-29 05:34:05.1774762445
Azuma's inequality: Expected sum of differences
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In some manner or other, you have to account for the case when $X_k - X_{k-1}$ is heavy tailed. Just consider the case $n=1$, and you can create counterexamples easily.