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 (or super-martingale) and

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

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

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

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

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

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

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

Contents

Simple example of Azuma's inequality for coin flips

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 X_i = \sum_{j=1}^i F_j yields a martingale with |Xk − Xk−1| ≤ 2, allowing us to apply Azuma's inequality. Specifically, we get

\Pr[X_N > t] \leq \exp\left(\frac{-t^2}{8 N}\right).

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.

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).

See also

References