@article{EJP147,
author = {Fuchang Gao and Jan Hannig and Fred Torcaso},
title = {Comparison Theorems for Small Deviations of Random Series},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {8},
year = {2003},
keywords = {small deviation, random series, bounded variation.},
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}.},
pages = {no. 21, 1-17},
issn = {1083-6489},
doi = {10.1214/EJP.v8-147},
url = {http://ejp.ejpecp.org/article/view/147}}