Hoeffding's inequality

Posted 2020-09-17

Let $\{Y_i: i\in J\}$ be zero mean independent complex-valued random variables satisfying $|Y_i|\le R.$ Then for all $c>0,$

$$P\left(|\sum_{i\in J}Y_i|>c\right)\le 4\exp\left(\frac{-c^2}{4R^2|J|}\right).$$

See, Hoeffding, W, Probability inequalities for sums of bounded random variables, Journal of the American Statistical Asociation, 58 (1963):13-30

or P. Shmerkin  Salem sets with no arithmetic progressions, international Mathematiics Research Notices.