Bernstein inequalities in probability theory

In probability theory, Bernstein inequalities give bounds on the probability that the sum of random variables deviates from its mean. In the simplest case, let X₁, ..., X_n be independent Bernoulli random variables taking values +1 and −1 with probability 1/2 (this distribution is also known as the Rademacher distribution), then for every positive ${\displaystyle \varepsilon }$,

Bernstein inequalities were proved and published by Sergei Bernstein in the 1920s and 1930s. Later, these inequalities were rediscovered several times in various forms. Thus, special cases of the Bernstein inequalities are also known as the Chernoff bound, Hoeffding's inequality and Azuma's inequality.

1. Let ${\displaystyle X_{1},\ldots ,X_{n}}$ be independent zero-mean random variables. Suppose that ${\displaystyle |X_{i}|\leq M}$ almost surely, for all ${\displaystyle i}$. Then, for all positive ${\displaystyle t}$,

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