Non-uniform Berry–Esseen bounds for martingales with applications to statistical estimation

Abstract

We establish nonuniform Berry-Esseen bounds for martingales under the conditional Bernstein condition. These bounds imply Cramér type large deviations for moderate x's, and are of exponential decay rate as de la Peña's inequality when x → ∞. Statistical applications associated with linear regressions and self-normalized large deviations are also provided.