Bernstein inequality

Bernstein inequalities give bounds on the probability that the sum of random variables deviates from its mean.

It is related to the (historically earliest) Bernstein inequalities, and to Hoeffding's inequality.

In mathematics, Bernstein inequality may refer to:

Bernstein inequalities were proved and published by Sergei Bernstein in the 1920s and 1930s.

Bernstein inequalities (probability theory)

Erdelyi started his career studying Markov and Bernstein inequalities for constrained polynomials in the late eighties.

A Bernstein inequalities (probability theory) was proved under weaker assumptions by Sergei Bernstein in 1937.

Bernstein inequalities / (F:R)

Thus, special cases of the Bernstein inequalities are also known as the Chernoff bound, Hoeffding's inequality and Azuma's inequality.

In the scalar setting, Bennett and Bernstein inequalities describe the upper tail of a sum of independent, zero-mean random variables that are either bounded or subexponential.

Hoeffding's inequality is a special case of the Azuma-Hoeffding inequality, and it is more general than the Bernstein inequality, proved by Sergei Bernstein in 1923.

The Bernstein inequality continues to hold for any (smooth) variety X. While the upper bound is an immediate consequence of the above interpretation of gr D in terms of the cotangent bundle, the lower bound is more subtle.

The Chernoff bound for a random variable X, which is the sum of n independent random variables , is obtained by applying e for some well-chosen value of t. This method was first applied by Sergei Bernstein to prove the related Bernstein inequalities.