Gaussian martingale inequality applies to random functions and maxima of empirical processes

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.