| Título: | Comparison Theorems for Small Deviations of Random Series |
| Autores: |
Gao, Fuchang; University of Idaho Hannig, Jan; Colorado State University Torcaso, Fred; Johns Hopkins University |
| Fecha: | 2003-01-01 |
| Publicador: | Electronic journal of probability |
| Fuente: |
 |
| Tipo: |
Peer-reviewed Article |
| Tema: |
small deviation, random series, bounded variation. primary 60F10; secondary 60G50 |
| Descripción: | 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}. |
| Idioma: | No aplica |
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