Cramér Type Moderate deviations for the Maximum of Self-normalized Sums

Zhishui Hu,Qi-Man Shao,Qiying Wang

doi:10.1214/ejp.v14-663

Abstract

Let $\{ X, X_i , i \geq 1\}$ be i.i.d. random variables, $S_k$ be the partial sum and $V_n^2 = \sum_{1\leq i\leq n} X_i^2$. Assume that $E(X)=0$ and $E(X^4) < \infty$. In this paper we discuss the moderate deviations of the maximum of the self-normalized sums. In particular, we prove that $P(\max_{1 \leq k \leq n} S_k \geq x V_n) / (1- \Phi(x)) \to 2$ uniformly in $x \in [0, o(n^{1/6}))$.

Highlights

Introduction and main resultsLet X, X1, X2, · · · be a sequence of i.i.d. random variables with mean zero
The past decade has witnessed a significant development on the limit theorems for the so-called self-normalized sum Sn/Vn
Shao (1997) showed that no moment conditions are needed for a self-normalized large deviation result P(Sn/Vn ≥ x n) and that the tail probability of Sn/Vn is Gaussian like when X1 is in the domain of attraction of the normal law and sub-Gaussian like when X is in the domain of attraction of a stable law, while Giné, Götze and Mason (1997) proved that the tails of Sn/Vn are uniformly sub-Gaussian when the sequence is stochastically bounded

Summary

Introduction

Introduction and main resultsLet X , X1, X2, · · · be a sequence of i.i.d. random variables with mean zero. Shao and Wang (2003) proved a Cramér type large deviation result (for independent random variables) under a Lindeberg type condition. Using a different approach from some known techniques for self-normalized sum, we establish a Cramér type large deviation result for the maximum of selfnormalized sums under a finite fourth moment.

Results

Conclusion