ProbabilityTheoryStatistics
Let be independent r.v.’s taking values in and have bounded differences such that then for any Indeed let be the canonical filtration and the Doob martingale. Next consider the martingale difference sequence so that we have the telescopic sum . Now define upper and lower bounds on : as and by the assumption of bounded differences, the result follows by application of Azuma’s Inequality. Note that Hoeffding’s Inequality is a special case with and .