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Self-normalized Cramér type moderate deviations for martingales and applications

Xiequan Fan, Qi-Man Shao

TL;DR

The paper develops self-normalized Cramér type moderate deviations for martingales under mild conditions, producing explicit non-asymptotic bounds for the tail relative error of the normal approximation of self-normalized sums $W_n = S_n/\sqrt{[S]_n}$. The main result is a flexible inequality that bounds $|\ln \frac{\mathbf{P}(W_n \ge x)}{1-\Phi(x)}|$ in terms of moment- and variance-control sequences, enabling uniform expansions for a wide range of $x$ and leading to a self-normalized MDP and Berry-Esseen-type bounds. The authors then demonstrate the utility of the theory through three applications: (i) self-normalized moderate deviations for the Student's $t$-statistic in martingale settings, (ii) stationary martingale difference sequences with a block-normalized statistic, and (iii) branching processes in a random environment, including a self-normalized $t$-type deviation for the Lotka–Nagaev estimator and an MDP for the corresponding sum. Collectively, the results extend prior work on martingale self-normalized deviations and provide practical inference tools for dependent data without requiring exponential moment assumptions.

Abstract

Cramér'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 Cramér 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 Cramér type moderate deviations for Student's $t$-statistic for branching processes in a random environment.

Self-normalized Cramér type moderate deviations for martingales and applications

TL;DR

The paper develops self-normalized Cramér type moderate deviations for martingales under mild conditions, producing explicit non-asymptotic bounds for the tail relative error of the normal approximation of self-normalized sums . The main result is a flexible inequality that bounds in terms of moment- and variance-control sequences, enabling uniform expansions for a wide range of and leading to a self-normalized MDP and Berry-Esseen-type bounds. The authors then demonstrate the utility of the theory through three applications: (i) self-normalized moderate deviations for the Student's -statistic in martingale settings, (ii) stationary martingale difference sequences with a block-normalized statistic, and (iii) branching processes in a random environment, including a self-normalized -type deviation for the Lotka–Nagaev estimator and an MDP for the corresponding sum. Collectively, the results extend prior work on martingale self-normalized deviations and provide practical inference tools for dependent data without requiring exponential moment assumptions.

Abstract

Cramér'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 Cramér 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 Cramér type moderate deviations for Student's -statistic for branching processes in a random environment.
Paper Structure (9 sections, 13 theorems, 123 equations)

This paper contains 9 sections, 13 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Applications
  4. Student's $t$-statistic
  5. Stationary martingale difference sequences
  6. Branching processes in a random environment
  7. Proofs of Theorems \ref{['thnds']} and \ref{['thndt']}
  8. Proof of Theorem \ref{['th000']}
  9. Proof of Theorem \ref{['th00s1']}

Key Result

Theorem 2.1

Assume that conditions (A1) and (A2) are satisfied. Then it holds uniformly for $0\leq x =o (\min\{ \gamma_n^{-1} , \delta_n^{-1} \})$. In particular, the last inequality implies that holds uniformly for $0\leq x =o(\min\{ \gamma_n^{-\rho/(2+\rho)} ,\, \delta_n^{-1/2} \} ).$ Moreover, the inequality (dfgs5sdf) remains valid when $\frac{\mathbf{P}(W_n \geq x)}{1-\Phi \left( x\right)}$ is replace

Theorems & Definitions (15)

  • Theorem 2.1
  • Remark 2.1
  • Corollary 2.1
  • Corollary 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.1
  • Theorem 3.3
  • Corollary 3.1
  • Theorem 3.4
  • ...and 5 more