  We investigate the properties of a discrete-time martingale $\{X_m\}_{m\in
\mathbb Z_{\geq 0}}$, where all differences between adjacent random variables
are limited to be not more than a constant as a promise. In this situation, it
is known that the Azuma-Hoeffding inequality holds, which gives an upper bound
of a probability for exceptional events. The inequality gives a simple form of
the upper bound, and it has been utilized for many investigations. However, the
inequality is not tight. We give an explicit expression of a tight upper bound,
and we show that it and the bound obtained from the Azuma-Hoeffding inequality
have different asymptotic behaviors.
