Limit Theorems for Self-Normalized Large Deviation

Abstract

Let $X, X_1, X_2, \cdots $ be i.i.d. random variables with zero mean and finite variance $\sigma^2$. It is well known that a finite exponential moment assumption is necessary to study limit theorems for large deviation for the standardized partial sums. In this paper, limit theorems for large deviation for self-normalized sums are derived only under finite moment conditions. In particular, we show that, if $EX^4<\infty$, then

$$\frac {P(S_n /V_n \geq x)}{1-\Phi(x)} = \exp\left\{ -\frac{x^3 EX^3}{3\sqrt{ n}\sigma^3} \right\} \left[ 1 + O\left(\frac{1+x}{\sqrt {n}}\right) \right], $$

for $x\ge 0$ and $x=O(n^{1/6})$, where $S_n=\sum_{i=1}^nX_i$ and $V_n= (\sum_{i=1}^n X_i^2)^{1/2}$.