Bernstein inequality

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Bernstein's inequality in probability theory is a more precise formulation of the classical Chebyshev inequality in probability theory, proposed by S.N. Bernshtein [1] in 1911; it permits one to estimate the probability of large deviations by a monotone decreasing exponential function (cf. Probability of large deviations). In fact, if the equations

hold for the independent random variables with

(where and is a constant independent of ), then the following inequality of Bernstein (where ) is valid for the sum :


where . For identically-distributed bounded random variables (, and , ) inequality (1) takes its simplest form:


where . A.N. Kolmogorov gave a lower estimate of the probability in (1). The Bernstein–Kolmogorov estimates are used, in particular, in proving the law of the iterated logarithm. Some idea of the accuracy of (2) may be obtained by comparing it with the approximate value of the left-hand side of (2) which is obtained by the central limit theorem in the form

where . Subsequent to 1967, Bernstein's inequalities were extended to include multi-dimensional and infinite-dimensional cases.


[1] S.N. Bernshtein, "Probability theory" , Moscow-Leningrad (1946) (In Russian)
[2] A.N. [A.N. Kolmogorov] Kolmogoroff, "Ueber das Gesetz des iterierten Logarithmus" Math. Ann. , 101 (1929) pp. 126–135
[3] W. Hoeffding, "Probability inequalities for sums of independent random variables" J. Amer. Statist. Assoc. , 58 (1963) pp. 13–30
[4] V.V. Yurinskii, "Exponential inequalities for sums of random vectors" J. Multivariate Anal. , 6 (1976) pp. 473–499

A.V. Prokhorov

Bernstein's inequality for the derivative of a trigonometric or algebraic polynomial gives an estimate of this derivative in terms of the polynomial itself. If is a trigonometric polynomial of degree not exceeding and if

then the following inequalities are valid for all (cf. [1]):

These estimates cannot be improved, since the number for

is sharp:

Bernstein's inequality for trigonometric polynomials is a special case of the following theorem [2]: If is an entire function of order and if

then one has

Bernstein's inequality for an algebraic polynomial has the following form [1]: If the polynomial

satifies the condition

then its derivative has the property

which cannot be improved. As was noted by S.N. Bernshtein [1], this inequality is a consequence of the proof of the Markov inequality given by A.A. Markov.

Bernstein's inequalities are in fact employed in proving converse theorems in the theory of approximation of functions. There are a number of generalizations of Bernstein's inequality, in particular for entire functions in several variables.


[1] S.N. [S.N. Bernshtein] Bernstein, "Sur l'ordre de la meilleure approximation des fonctions continues par des polynômes" Acad. R. Belgique, Cl. Sci. Mém. Coll. 4. Sér. II , 4 (1922)
[2] S.N. [S.N. Bernshtein] Bernstein, "Sur une propriété des fonctions entières" C.R. Acad. Sci. Paris , 176 (1923) pp. 1603–1605
[3] S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)

N.P. KorneichukV.P. Motornyi



[a1] I.P. Natanson, "Constructive function theory" , 1–3 , F. Ungar (1964–1965) (Translated from Russian)
[a2] G.G. Lorentz, "Approximation of functions" , Holt, Rinehart & Winston (1966) pp. Chapt. 2
How to Cite This Entry:
Bernstein inequality. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.V. Prokhorov, N.P. Korneichuk, V.P. Motornyi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article