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Article Dans Une Revue Statistics Année : 2017

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.

Dates et versions

hal-01449001 , version 1 (29-01-2017)

Identifiants

Citer

Xiequan Fan, Ion Grama, Quansheng Liu. Martingale inequalities of type Dzhaparidze and van Zanten . Statistics, 2017, 51 (6), pp.1200-1213. ⟨10.1080/02331888.2017.1318138⟩. ⟨hal-01449001⟩
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