  We obtain a Bernstein type Gaussian concentration inequality for martingales.
Our inequality improves the Azuma-Hoeffding inequality for moderate deviations
$x$. Following the work of McDiarmid (1989), Talagrand (1996) and Boucheron,
Lugosi and Massart (2000,2003), we show that our result can be applied to the
concentration of random functions, Erd\"{o}s-R\'{e}nyi random graph, and maxima
of empirical processes. Several interesting Gaussian concentration inequalities
have been obtained.
