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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 theconcentration of random functions, Erd\"{o}s-R\'{e}nyi random graph, and maximaof empirical processes. Several interesting Gaussian concentration inequalitieshave been obtained.

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