Self-normalized Cram\'{e}r type moderate deviations for martingales and applications

TL;DR

This paper establishes self-normalized Cramér type moderate deviations for martingales under mild conditions, extending previous work and applying the results to Student's t-statistic, stationary martingale differences, and branching processes in random environments.

Contribution

It introduces new self-normalized Cramér type moderate deviation results for martingales, broadening their applicability in statistical inference and stochastic process analysis.

Findings

Extended moderate deviation results to martingales under mild conditions.

Applied results to Student's t-statistic in branching processes.

Provided theoretical justifications for statistical estimators.

Abstract

Cram\'er's moderate deviations give a quantitative estimate for the relative error of the normal approximation and provide theoretical justifications for many estimator used in statistics. In this paper, we establish self-normalized Cram\'{e}r type moderate deviations for martingales under some mile conditions. The result extends an earlier work of Fan, Grama, Liu and Shao [Bernoulli, 2019]. Moreover, applications of our result to Student's statistic, stationary martingale difference sequences and branching processes in a random environment are also discussed. In particular, we establish Cram\'{e}r type moderate deviations for Student's $t$-statistic for branching processes in a random environment.