Limit theorems for self-normalized large deviation

Research Report SRR02-002

Limit theorems for self-normalized large deviation

Qiying Wang

Abstract: Let X, X₁, X₂, ... be i.i.d. random variables with zero mean and finite variance \sigma². 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 a finite third moment. In particular, we show that, if E|X|³< \infty, then

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

for x\geq 0 and x=O(n^1/6), where S_n=\sum_{i=1}^{n} X_i and V_n=(\sum_{i=1}^{n} X_i²)^1/2.

Primary AMS Classification: 60F05
Date: 29 April 2002

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