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 ,

Bernstein inequalities were proven 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. The martingale case of the Bernstein inequality is known as Freedman's inequality [5] and its refinement is known as Hoeffding's inequality.[6]

Some of the inequalities

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

 

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

 

Then

 

3. Let   be independent zero-mean random variables. Suppose that

 

for all integer   Denote

 

Then,

 

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

 

Then

 

More general results for martingales can be found in Fan et al. (2015).[7]

Proofs

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

 

for a suitable choice of the parameter  .

Generalizations

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The Bernstein inequality can be generalized to Gaussian random matrices. Let   be a scalar where   is a complex Hermitian matrix and   is complex vector of size  . The vector   is a Gaussian vector of size  . Then for any  , we have

 

where   is the vectorization operation and   where   is the largest eigenvalue of  . The proof is detailed here.[8] Another similar inequality is formulated as

 

where  .

See also

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References

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  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. ^ Freedman, D.A. (1975). "On tail probabilities for martingales". Ann. Probab. 3: 100–118.
  6. ^ Fan, X.; Grama, I.; Liu, Q. (2012). "Hoeffding's inequality for supermartingales". Stochastic Process. Appl. 122: 3545–3559.
  7. ^ Fan, X.; Grama, I.; Liu, Q. (2015). "Exponential inequalities for martingales with applications". Electronic Journal of Probability. 20. Electron. J. Probab. 20: 1–22. arXiv:1311.6273. doi:10.1214/EJP.v20-3496. S2CID 119713171.
  8. ^ Ikhlef, Bechar (2009). "A Bernstein-type inequality for stochastic processes of quadratic forms of Gaussian variables". arXiv:0909.3595 [math.ST].

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

A modern translation of some of these results can also be found in Prokhorov, A.V.; Korneichuk, N.P.; Motornyi, V.P. (2001) [1994], "Bernstein inequality", Encyclopedia of Mathematics, EMS Press