Martingale inequalities of type Dzhaparidze and van Zanten
Résumé
Freedman’s inequality is a supermartingale counterpart to Bennett’s inequality. This re- sult shows that the tail probabilities of a supermartingale is controlled by the quadratic characteristic and a uniform upper bound for the supermartingale difference sequence. Re- placing the quadratic characteristic by Hyk := ki=1 E(ξi2|Fi−1) + ξi21{|ξi|>y} , Dzhaparidze and van Zanten (Stochastic Process. Appl., 2001) have extended Freedman’s inequality to martingales with unbounded differences. In this paper, we prove that Hyk can be refined to Gyk := ki=1 E(ξi21{ξi≤y}|Fi−1)+ξi21{ξi>y} . Moreover, we also establish two inequalities of type Dzhaparidze and van Zanten. These results extend Sason’s inequality (Statist. Probab. Lett., 2012) to the martingales with possibly unbounded differences and establish the con- nection between Sason’s inequality and De la Pen ̃a’s inequality (Ann. Probab., 1999). An application to self-normalized deviations is given.