Azuma's inequality

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In probability theory, the Azuma–Hoeffding inequality (named after Kazuoki Azuma and Wassily Hoeffding) gives a concentration result for the values of martingales that have bounded differences.

Suppose { Xk : k = 0, 1, 2, 3, ... } is a martingale (or super-martingale) and

almost surely. Then for all positive integers N and all positive reals t,

And symmetrically (when Xk is a sub-martingale):

If X is a martingale, using both inequalities above and applying the union bound allows one to obtain a two-sided bound:

Azuma's inequality applied to the Doob martingale gives McDiarmid's inequality which is common in the analysis of randomized algorithms.

Simple example of Azuma's inequality for coin flips[edit]

Let Fi be a sequence of independent and identically distributed random coin flips (i.e., let Fi be equally likely to be −1 or 1 independent of the other values of Fi). Defining yields a martingale with |Xk − Xk−1| ≤ 1, allowing us to apply Azuma's inequality. Specifically, we get

For example, if we set t proportional to n, then this tells us that although the maximum possible value of Xn scales linearly with n, the probability that the sum scales linearly with n decreases exponentially fast with n.

If we set we get:

which means that the probability of deviating more than approaches 0 as n goes to infinity.


A similar inequality was proved under weaker assumptions by Sergei Bernstein in 1937.

Hoeffding proved this result for independent variables rather than martingale differences, and also observed that slight modifications of his argument establish the result for martingale differences (see page 18 of his 1963 paper).

See also[edit]


  • Alon, N.; Spencer, J. (1992). The Probabilistic Method. New York: Wiley.
  • Azuma, K. (1967). "Weighted Sums of Certain Dependent Random Variables" (PDF). Tôhoku Mathematical Journal. 19 (3): 357–367. doi:10.2748/tmj/1178243286. MR 0221571.
  • Bernstein, Sergei N. (1937). На определенных модификациях неравенства Чебишева [On certain modifications of Chebyshev's inequality]. Doklady Akademii Nauk SSSR (in Russian). 17 (6): 275–277. (vol. 4, item 22 in the collected works)
  • McDiarmid, C. (1989). "On the method of bounded differences". Surveys in Combinatorics. London Math. Soc. Lectures Notes 141. Cambridge: Cambridge Univ. Press. pp. 148–188. MR 1036755.
  • Hoeffding, W. (1963). "Probability inequalities for sums of bounded random variables". Journal of the American Statistical Association. 58 (301): 13–30. doi:10.2307/2282952. JSTOR 2282952. MR 0144363.
  • Godbole, A. P.; Hitczenko, P. (1998). Beyond the method of bounded differences. DIMACS Series in Discrete Mathematics and Theoretical Computer Science. 41. pp. 43–58. doi:10.1090/dimacs/041/03. ISBN 9780821808276. MR 1630408.