Comparison theorems for small deviations of weighted series (Q2905816)

The aim of the present paper is to extend the following comparison result of \textit{W. V. Li} [J. Theor. Probab. 5, No. 1, 1--31 (1992; Zbl 0743.60009)]. Let \(\{a_n\}\) and \(\{b_n\}\) be two summable sequences of positive numbers such that NEWLINE\[NEWLINE\sum_{n=1}^\infty \left|1-\frac{a_n}{b_n}\right|<\infty. NEWLINE\]NEWLINE If \(\{Z_n\}\) are i.i.d. standard normal, then as \(\epsilon\to 0\), it holds NEWLINE\[NEWLINE \operatorname{P}\left(\sum_{n=1}^\infty a_n Z_n^2<\epsilon^2\right) \sim \left(\prod_{n=1}^\infty\frac{b_n}{a_n}\right)^{1/2}\, \operatorname{P}\left(\sum_{n=1}^\infty b_n Z_n^2<\epsilon^2\right)\,. NEWLINE\]NEWLINE For example, in the present paper, the following generalization of this result is proved. Let \(X,X_1,X_2,\dots\) be a sequence of i.i.d. non-negative random variables such that \(X\) is in the domain of attraction of a stable law with index greater than one, and let its distribution function be regularly varying at zero with index \(\beta>0\). If the \(\{a_n\}\) and \(\{b_n\}\) are as before, then, if \(\epsilon\to 0\), it follows NEWLINE\[NEWLINE \operatorname{P}\left(\sum_{n=1}^\infty a_n X_n<\epsilon\right) \sim \left(\prod_{n=1}^\infty\frac{b_n}{a_n}\right)^{\beta} \operatorname{P}\left(\sum_{n=1}^\infty b_n X_n<\epsilon\right). NEWLINE\]