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Limit Theorems for Self-Normalized Large Deviation


 
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1. Title Title of document Limit Theorems for Self-Normalized Large Deviation
 
2. Creator Author's name, affiliation, country Qiying Wang; University of Sydney
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Cram'er large deviation, limit theorem,
 
3. Subject Subject classification Primary 60F05, Secondary 62E20.
 
4. Description 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 $V_n= (\sum_{i=1}^n X_i^2)^{1/2}$.
 
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7. Date (YYYY-MM-DD) 2005-11-14
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/289
 
10. Identifier Digital Object Identifier 10.1214/EJP.v10-289
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 10
 
12. Language English=en
 
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