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Comparison Theorems for Small Deviations of Random Series
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Fuchang Gao, University of Idaho
Jan Hannig, Colorado State University
Fred Torcaso, Johns Hopkins University
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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
$$Prleft(sum_{n=1}^{infty}a_nxi_nleq epsright)sim
left(prod_{n=1}^{infty}frac{b_n}{a_n}right)^{-alpha}
Prleft(sum_{n=1}^{infty}b_nxi_nleq epsright),$$ where
$${ alpha=lim_{xto infty}frac{log F(1/x)}{log x}}< 0$$
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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Full text: PDF
Pages: 1-17
Published on: December 27, 2003
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Electronic Journal of Probability. ISSN: 1083-6489 |
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