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


 * $$|X_k - X_{k-1}| < c_k,$$

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


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

Applying Azuma's inequality to the martingale -X 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.

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 &minus;1 independent of the other values of $$F_i$$). Defining $$X_i = \sum_{j=1}^i F_j$$ yields a martingale with $$|X_{k}-X_{k-1}|\leq 1$$, allowing us to apply Azuma's inequality. Specifically, we get
 * $$ \Pr[X_N > X_0 + t] \leq \exp\left(\frac{-t^2}{2 N}\right).$$

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's result was derived in the special case of sums of independent random variables rather than martingales.