Cram\'er large deviation expansions for martingales under Bernstein's condition

TL;DR

This paper derives optimal large deviation probability expansions for martingales with differences satisfying Bernstein's condition, extending classical Cramér's results using conjugate distribution techniques.

Contribution

It provides new, optimal large deviation expansions for martingales under Bernstein's condition, enhancing understanding of their tail behaviors.

Findings

Expansions are of the same order as classical Cramér's results.

Results are optimal for martingales with Bernstein's condition.

Uses conjugate distribution technique for derivation.

Abstract

By using the conjugate distribution technique of Cram\'er, we obtain some expansions of large deviation probabilities for martingales with differences satisfying the conditional Bernstein's condition. The expansions are of the same order as in the classical Cram\'er's large deviation result and are therefore optimal.