The Annals of Probability

A Ratio Limit Theorem for the Tails of Weighted Sums

Holger Rootzen

Abstract

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.

Article information

Source
Ann. Probab., Volume 15, Number 2 (1987), 728-747.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176992168

Digital Object Identifier
doi:10.1214/aop/1176992168

Mathematical Reviews number (MathSciNet)
MR885140

Zentralblatt MATH identifier
0637.60025

JSTOR