Bernstein inequalities (probability theory)

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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 ,

Bernstein inequalities were proved and published by Sergei Bernstein in the 1920s and 1930s.[1][2][3][4] 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.

Some of the inequalities[edit]

1. Let be independent zero-mean random variables. Suppose that almost surely, for all . Then, for all positive ,

2. Let be independent random variables. Suppose that for some positive real and every integer ,


3. Let be independent random variables. Suppose that

for all integer . Denote


4. Bernstein also proved generalizations of the inequalities above to weakly dependent random variables. For example, inequality (2) can be extended as follows. be possibly non-independent random variables. Suppose that for all integer ,


More general results for martingales can be found in Fan et al. (2015).[5]


The proofs are based on an application of Markov's inequality to the random variable

for a suitable choice of the parameter .

See also[edit]


(according to: S.N.Bernstein, Collected Works, Nauka, 1964)

  1. ^ S.N.Bernstein, "On a modification of Chebyshev’s inequality and of the error formula of Laplace" vol. 4, #5 (original publication: Ann. Sci. Inst. Sav. Ukraine, Sect. Math. 1, 1924)
  2. ^ Bernstein, S. N. (1937). "Об определенных модификациях неравенства Чебышева" [On certain modifications of Chebyshev's inequality]. Doklady Akademii Nauk SSSR. 17 (6): 275–277. 
  3. ^ S.N.Bernstein, "Theory of Probability" (Russian), Moscow, 1927
  4. ^ J.V.Uspensky, "Introduction to Mathematical Probability", McGraw-Hill Book Company, 1937
  5. ^ Fan, X.; Grama, I.; Liu, Q. (2015). "Exponential inequalities for martingales with applications". Electron. J. Probab. 20: 1–22. doi:10.1214/EJP.v20-3496. 

A modern translation of some of these results can also be found in Prokhorov, A.V.; Korneichuk, N.P. (2001) [1994], "Bernstein inequality", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4