Bernstein inequalities (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, ..., Xn 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 ε {\displaystyle \varepsilon } , P ( | 1 n ∑ i = 1 n X i | > ε ) ≤ 2 exp ⁡ ( − n ε 2 2 ( 1 + ε 3 ) ) .

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