Definitions

# Azuma's inequality

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 and

$|X_k - X_\left\{k-1\right\}| < c_k,$

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

$P\left(X_N - X_0 geq t\right) leq expleft \left(\left\{-t^2 over 2 sum_\left\{k=1\right\}^\left\{N\right\}c_k^2\right\} right\right).$

Applying Azuma's inequality to the martingale -X and applying the union bound allows one to obtain a two-sided bound:

$P\left(|X_N - X_0| geq t\right) leq 2expleft \left(\left\{-t^2 over 2 sum_\left\{k=1\right\}^\left\{N\right\}c_k^2\right\} right\right).$

Azuma's inequality applied to the Doob martingale gives the method of bounded differences (MOBD) which is common in the analysis of randomized algorithms.

## Simple example of Azuma's inequality for coin flips

Let $F_i$ be a sequence of independent and identically distributed random coin flips (i.e., let $F_i$ be equally like to be +1 or −1 independent of the other values of $F_i$). Defining $X_i = sum_\left\{j=1\right\}^i F_j$ yields a martingale with $|X_\left\{k\right\}-X_\left\{k-1\right\}|leq 1$, allowing us to apply Azuma's inequality. Specifically, we get
$Pr\left[X_N > X_0 + t\right] leq expleft\left(frac\left\{-t^2\right\}\left\{2 N\right\}right\right).$

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

## Remark

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).

## References

• N. Alon & J. Spencer, The Probabilistic Method. Wiley, New York 1992.
• K. Azuma, "Weighted Sums of Certain Dependent Random Variables". Tôhoku Math. Journ. 19, 357-367 (1967)
• S.N. Bernstein, "On certain modifications of Chebyshev's inequality", Doklady Akad. Nauk SSSR, 17, n. 6, (1937), 275-277 (vol. 4, item 22 in the collected works)
• C. McDiarmid, On the method of bounded differences. In Surveys in Combinatorics, London Math. Soc. Lectures Notes 141, Cambridge Univ. Press, Cambridge 1989, 148-188.
• W. Hoeffding, "Probability inequalities for sums of bounded random variables", J. Amer. Statist. Assoc. 58, 13-30, 1963 MR0144363
• A. P. Godbole and P. Hitczenko, "Beyond the method of bounded differences", DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 41, 43-58, 1998 MR1630408

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