Nonuniform Berry-Esseen bounds for martingales with applications to statistical estimation

We establish nonuniform Berry-Esseen bounds for martingales under the conditional Bernstein condition. These bounds imply Cram'er type large deviations for moderate $x$’s, and are of exponential decay rate as de la Pe~na’s inequality when $x\to \mathrm{\infty }$. Statistical applications associated with linear regressions and self-normalized large deviations are also provided.