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Comparison Theorems for Small Deviations of Random Series


 
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1. Title Title of document Comparison Theorems for Small Deviations of Random Series
 
2. Creator Author's name, affiliation, country Fuchang Gao; University of Idaho
 
2. Creator Author's name, affiliation, country Jan Hannig; Colorado State University
 
2. Creator Author's name, affiliation, country Fred Torcaso; Johns Hopkins University
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) small deviation, random series, bounded variation.
 
3. Subject Subject classification primary 60F10; secondary 60G50
 
4. Description Abstract Let ${\xi_n}$ be a sequence of i.i.d. positive random variables with common distribution function $F(x)$. Let ${a_n}$ and ${b_n}$ be two positive non-increasing summable sequences such that ${\prod_{n=1}^{\infty}(a_n/b_n)}$ converges. Under some mild assumptions on $F$, we prove the following comparison $$P\left(\sum_{n=1}^{\infty}a_n \xi_n \leq \varepsilon \right) \sim \left(\prod_{n=1}^{\infty}\frac{b_n}{a_n}\right)^{-\alpha} P \left(\sum_{n=1}^{\infty}b_n \xi_n \leq \varepsilon \right),$$ where $${ \alpha=\lim_{x\to \infty}\frac{\log F(1/x)}{\log x}}< 0$$ is the index of variation of $F(1/\cdot)$. When applied to the case $\xi_n=|Z_n|^p$, where $Z_n$ are independent standard Gaussian random variables, it affirms a conjecture of Li cite {Li1992a}.
 
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7. Date (YYYY-MM-DD) 2003-12-27
 
8. Type Status & genre Peer-reviewed Article
 
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9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/147
 
10. Identifier Digital Object Identifier 10.1214/EJP.v8-147
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 8
 
12. Language English=en
 
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