Concentration inequality

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In probability theory, concentration inequalities provide mathematical bounds on the probability of a random variable deviating from some value (typically, its expected value).

The law of large numbers of classical probability theory states that sums of independent random variables, under mild conditions, concentrate around their expectation with a high probability. Such sums are the most basic examples of random variables concentrated around their mean.

Concentration inequalities can be sorted according to how much information about the random variable is needed in order to use them.[citation needed]

Markov's inequality[edit]

Let be a random variable that is non-negative (almost surely). Then, for every constant ,

Note the following extension to Markov's inequality: if is a strictly increasing and non-negative function, then

Chebyshev's inequality[edit]

Chebyshev's inequality requires the following information on a random variable :

  • The expected value is finite.
  • The variance is finite.

Then, for every constant ,

or equivalently,

where is the standard deviation of .

Chebyshev's inequality can be seen as a special case of the generalized Markov's inequality applied to the random variable with .

Vysochanskij–Petunin inequality[edit]

Let X be a random variable with unimodal distribution, mean μ and finite, non-zero variance σ2. Then, for any

(For a relatively elementary proof see e.g.[1]).

One-sided Vysochanskij–Petunin inequality[edit]

For a unimodal random variable and , the one-sided Vysochanskij-Petunin inequality[2] holds as follows:

Paley–Zygmund inequality[edit]

In contrast to most commonly used concentration inequalities, the Paley-Zygmund inequality provides a lower bound on the deviation probability.

Cantelli's inequality[edit]

Gauss's inequality[edit]

Chernoff bounds[edit]

The generic Chernoff bound[3]: 63–65  requires the moment generating function of , defined as It always exists, but may be infinite. From Markov's inequality, for every :

and for every :

There are various Chernoff bounds for different distributions and different values of the parameter . See [4]: 5–7  for a compilation of more concentration inequalities.

Bounds on sums of independent bounded variables[edit]

Let be independent random variables such that, for all i:

almost surely.

Let be their sum, its expected value and its variance:

It is often interesting to bound the difference between the sum and its expected value. Several inequalities can be used.

1. Hoeffding's inequality says that:

2. The random variable is a special case of a martingale, and . Hence, the general form of Azuma's inequality can also be used and it yields a similar bound:

This is a generalization of Hoeffding's since it can handle other types of martingales, as well as supermartingales and submartingales. See Fan et al. (2015).[5] Note that if the simpler form of Azuma's inequality is used, the exponent in the bound is worse by a factor of 4.

3. The sum function, , is a special case of a function of n variables. This function changes in a bounded way: if variable i is changed, the value of f changes by at most . Hence, McDiarmid's inequality can also be used and it yields a similar bound:

This is a different generalization of Hoeffding's since it can handle other functions besides the sum function, as long as they change in a bounded way.

4. Bennett's inequality offers some improvement over Hoeffding's when the variances of the summands are small compared to their almost-sure bounds C. It says that:


5. The first of Bernstein's inequalities says that:

This is a generalization of Hoeffding's since it can handle random variables with not only almost-sure bound but both almost-sure bound and variance bound.

6. Chernoff bounds have a particularly simple form in the case of sum of independent variables, since .

For example,[6] suppose the variables satisfy , for . Then we have lower tail inequality:

If satisfies , we have upper tail inequality:

If are i.i.d., and is the variance of , a typical version of Chernoff inequality is:

7. Similar bounds can be found in: Rademacher distribution#Bounds on sums

Efron–Stein inequality[edit]

The Efron–Stein inequality (or influence inequality, or MG bound on variance) bounds the variance of a general function.

Suppose that , are independent with and having the same distribution for all .

Let Then

A proof may be found in e.g.,.[7]

Bretagnolle–Huber–Carol inequality[edit]

Bretagnolle–Huber–Carol Inequality bounds the difference between a vector of multinomially distributed random variables and a vector of expected values.[8][9] A simple proof appears in [10](Appendix Section).

If a random vector is multinomially distributed with parameters and satisfies then

This inequality is used to bound the total variation distance.

Mason and van Zwet inequality[edit]

The Mason and van Zwet inequality[11] for multinomial random vectors concerns a slight modification of the classical chi-square statistic.

Let the random vector be multinomially distributed with parameters and such that for Then for every and there exist constants such that for all and satisfying and we have

Dvoretzky–Kiefer–Wolfowitz inequality[edit]

The Dvoretzky–Kiefer–Wolfowitz inequality bounds the difference between the real and the empirical cumulative distribution function.

Given a natural number , let be real-valued independent and identically distributed random variables with cumulative distribution function F(·). Let denote the associated empirical distribution function defined by

So is the probability that a single random variable is smaller than , and is the average number of random variables that are smaller than .


Anti-concentration inequalities[edit]

Anti-concentration inequalities, on the other hand, provide an upper bound on how much a random variable can concentrate around a quantity.

For example, Rao and Yehudayoff[12] show that there exists some such that, for most directions of the hypercube , the following is true:[clarification needed]

where is drawn uniformly from a subset of large enough size.

Such inequalities are of importance in several fields, including communication complexity (e.g., in proofs of the gap Hamming problem[13]) and graph theory.[14]

An interesting anti-concentration inequality for weighted sums of independent Rademacher random variables can be obtained using the Paley–Zygmund and the Khintchine inequalities.[15]


  1. ^ Pukelsheim, F., 1994. The Three Sigma Rule. The American Statistician, 48(2), pp. 88–91
  2. ^ Mercadier, Mathieu; Strobel, Frank (2021-11-16). "A one-sided Vysochanskii-Petunin inequality with financial applications". European Journal of Operational Research. 295 (1): 374–377. doi:10.1016/j.ejor.2021.02.041. ISSN 0377-2217.
  3. ^ Mitzenmacher, Michael; Upfal, Eli (2005). Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press. ISBN 0-521-83540-2.
  4. ^ Slagle, N.P. (2012). "One Hundred Statistics and Probability Inequalities". arXiv:2102.07234.
  5. ^ 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.
  6. ^ Chung, Fan; Lu, Linyuan (2010). "Old and new concentration inequalities" (PDF). Complex Graphs and Networks. American Mathematical Society. Retrieved August 14, 2018.
  7. ^ Boucheron, St{\'e}phane; Lugosi, G{\'a}bor; Bousquet, Olivier (2004). "Concentration inequalities". Advanced Lectures on Machine Learning: ML Summer Schools 2003, Canberra, Australia, February 2–14, 2003, T{\"u}bingen, Germany, August 4–16, 2003, Revised Lectures. Springer: 208–240.
  8. ^ Bretagnolle, Jean; Huber-Carol, Catherine (1978). Lois empiriques et distance de Prokhorov. Lecture Notes in Mathematics. Vol. 649. pp. 332–341. doi:10.1007/BFb0064609. ISBN 978-3-540-08761-8.
  9. ^ van der Vaart, A.W.; Wellner, J.A. (1996). Weak convergence and empirical processes: With applications to statistics. Springer Science & Business Media.
  10. ^ Yuto Ushioda; Masato Tanaka; Tomomi Matsui (2022). "Monte Carlo Methods for the Shapley–Shubik Power Index". Games. 13 (3): 44. arXiv:2101.02841. doi:10.3390/g13030044.
  11. ^ Mason, David M.; Willem R. Van Zwet (1987). "A Refinement of the KMT Inequality for the Uniform Empirical Process". The Annals of Probability. 15 (3): 871–884. doi:10.1214/aop/1176992070.
  12. ^ Rao, Anup; Yehudayoff, Amir (2018). "Anti-concentration in most directions". Electronic Colloquium on Computational Complexity.
  13. ^ Sherstov, Alexander A. (2012). "The Communication Complexity of Gap Hamming Distance". Theory of Computing.
  14. ^ Matthew Kwan; Benny Sudakov; Tuan Tran (2018). "Anticoncentration for subgraph statistics". Journal of the London Mathematical Society. 99 (3): 757–777. arXiv:1807.05202. Bibcode:2018arXiv180705202K. doi:10.1112/jlms.12192. S2CID 54065186.
  15. ^ Veraar, Mark (2009). "On Khintchine inequalities with a weight". arXiv:0909.2586v1 [math.PR].

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