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 $\textrm{H}_k^y:= \sum_{i=1}^k\left(\mathbf{E}(\xi_i^2 |\mathcal{F}_{i-1}) +\xi_i^2\textbf{1}_{\{|\xi_i|> y\}}\right),$ 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 $\textrm{H}_k^y$ can be refined to $\textrm{G}_k^{y} :=\sum_{i=1}^k \left( \mathbf{E}(\xi_i^2\textbf{1}_{\{\xi_i \leq y\}} |\mathcal{F}_{i-1}) + \xi_i^2\textbf{1}_{\{\xi_i> y\}}\right).$ 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 Pe\~{n}a's inequality (\emph{Ann.\ Probab.,} 1999). An application to self-normalized deviations is given.