Martingale inequalities of type Dzhaparidze and van Zanten

Abstract

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 ${\text{H}}_k^y:={\sum }_{i=1}^k\left(\mathbf{E}({\xi }_i^2|{\mathcal{F}}_{i-1})+{\xi }_i^2{\text{1}}_{\{|{\xi }_i|>y\}}\right),$ Dzhaparidze and van Zanten (\emph{Stochastic Process. Appl.}, 2001) have extended Freedman's inequality to martingales with unbounded differences. In this paper, we prove that ${\text{H}}_k^y$ can be refined to ${\text{G}}_k^y:={\sum }_{i=1}^k\left(\mathbf{E}({\xi }_i^2{\text{1}}_{\{{\xi }_i\le y\}}|{\mathcal{F}}_{i-1})+{\xi }_i^2{\text{1}}_{\{{\xi }_i>y\}}\right).$ Moreover, we also establish two inequalities of type Dzhaparidze and van Zanten. These results extend Sason's inequality (\emph{Statist. Probab. Lett.}, 2012) to the martingales with possibly unbounded differences and establish the connection between Sason's inequality and De la Peña's inequality (\emph{Ann.\ Probab.,} 1999). An application to self-normalized deviations is given.

References in corpus (1)

• Exponential inequalities for self-normalized martingales with applications

Cited by in corpus (1)

• Bound on the maximal function associated to the bounded law of the iterated logarithms via orthomartingale approximation