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 $${ \alpha=\lim_{x\to \infty}\frac{\log F(1/x)}{\log x}} \lt 0$$ 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 (1992).
Citation
Fuchang Gao. Jan Hannig. Fred Torcaso. "Comparison Theorems for Small Deviations of Random Series." Electron. J. Probab. 8 1 - 17, 2003. https://doi.org/10.1214/EJP.v8-147
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