Xiequan Fan

Cramér large deviation expansions for martingales under Bernstein's condition

An expansion of large deviation probabilities for martingales is given, which extends the classical result due to Cramer to the case of martingale differences satisfying the conditional Bernstein condition. The upper bound of the range of validity and the remainder of our expansion is the same as in the Cramer result and therefore are optimal. Our result implies a moderate deviation principle for martingales.

Martingale inequalities of type Dzhaparidze and van Zanten

ABSTRACT Freedmans inequality is a supermartingale counterpart to Bennetts 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 Freedmans inequality to martingales with unbounded differences. In this paper, we prove that can be refined to . Moreover, we also establish two inequalities of type Dzhaparidze and van Zanten. These results extend Sasons 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 Sasons inequality and De la Peñas 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.