Martingale inequalities of type Dzhaparidze and van Zanten.

Freedman's inequality is a supermartingale counterpart to Bennett's inequality. This result shows that the tail probabilities of a supermartingale is controlled by the quadratic characteristic and a uniform upper bound for the supermartingale difference sequence. Replacing the quadratic characteristic by, Dzhaparidze and van Zanten [On Bernstein-type inequalities for martingales. Stoch Process Appl. 2001;93:109–117] have extended Freedman's inequality to martingales with unbounded differences. In this paper, we prove thatcan be refined to. Moreover, we also establish two inequalities of type Dzhaparidze and van Zanten. These results extend Sason's inequality [Tightened exponential bounds for discrete-time conditionally symmetric martingales with bounded jumps. Statist Probab Lett. 2013;83:1928–1936] to martingales with possibly unbounded differences and establish the connection between Sason's inequality and De la Peña's inequality [A general class of exponential inequalities for martingales and ratios. Ann Probab. 1999;27(1):537–564]. An application to self-normalized deviations is given. [ABSTRACT FROM AUTHOR]