Open Access
Translator Disclaimer
April, 1987 A Ratio Limit Theorem for the Tails of Weighted Sums
Holger Rootzen
Ann. Probab. 15(2): 728-747 (April, 1987). DOI: 10.1214/aop/1176992168


Let $\{Z_\lambda; \lambda = 0, \pm 1,\ldots\}$ be i.i.d. random variables which have a density $f$ which satisfies $f(z) \sim Kz^\alpha\exp\{-z^p\}$ as $z \rightarrow \infty$ for some constants $p > 1, K > 0$, and $\alpha$. Further let $q$ be defined by $p^{-1} + q^{-1} = 1$, and let $\{c_\lambda\}$ be constants with $c_\lambda = O(|\lambda|^{-\theta})$ for some $\theta > \max\{1,2/q\}$. Then, e.g., if $f$ is symmetric $\frac{P(\sum c_\lambda Z_\lambda > z + x/z^{p/q})}{P(\sum c_\lambda Z_\lambda > z)} \rightarrow \exp \{-p\|c\|^{-p}_q x\}, \text{as} z \rightarrow \infty$, for $\|c\|_q = (\sum|c_\lambda|^q)^{1/q}$, and similar results are obtained also for nonsymmetric cases, under some mild further smoothness restrictions. In addition, an order bound for $P(\sum c_\lambda Z_\lambda > z)$ itself is obtained, and precise estimates of this quantity are found for the special case of finite sums. In the companion paper [7], the results are crucially used to study extreme values of moving average processes.


Download Citation

Holger Rootzen. "A Ratio Limit Theorem for the Tails of Weighted Sums." Ann. Probab. 15 (2) 728 - 747, April, 1987.


Published: April, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0637.60025
MathSciNet: MR885140
Digital Object Identifier: 10.1214/aop/1176992168

Primary: 60E99

Keywords: large deviations , tails of convolutions , weighted sums

Rights: Copyright © 1987 Institute of Mathematical Statistics


Vol.15 • No. 2 • April, 1987
Back to Top