  The Bernstein inequality is a tight upper bound on tail probabilities for
independent random variables. Freedman extended the Bernstein inequality to
martingales with differences bounded from above, and then Dzhaparidze and van
Zanten generalized Freedman's result to non-bounded locally square integrable
martingales. In this paper, we derive some Dzhaparidze-van Zanten type
inequalities for self-normalized martingales with square integrable and
non-square integrable differences.
