Open Access
Translator Disclaimer
April, 1986 Extreme Value Theory for Moving Average Processes
Holger Rootzen
Ann. Probab. 14(2): 612-652 (April, 1986). DOI: 10.1214/aop/1176992534

Abstract

This paper studies extreme values in infinite moving average processes $X_t = \sum_\lambda c_{\lambda - t} Z_\lambda$ defined from an i.i.d. noise sequence $\{Z_\lambda\}$. In particular this includes the ARMA-processes often used in time series analysis. A fairly complete extremal theory is developed for the cases when the d.f. of the $Z_\lambda$'s has a smooth tail which decreases approximately as $\exp\{- z^p\}$ as $z \rightarrow \infty$, for $0 < p < \infty$, or as a power of $z$. The influence of the averaging on extreme values depends on $p$ and the $c_\lambda$'s in a rather intricate way. For $p = 2$, which includes normal sequences, the correlation function $r_t = \sum_\lambda c_{\lambda - t}c_\lambda/\sum_\lambda c^2_\lambda$ determines extremal behavior while, perhaps more surprisingly, for $p \neq 2$ correlations have little bearing on extremes. Further, the sample paths of $\{X_t\}$ near extreme values asymptotically assume a specific nonrandom form, which again depends on $p$ and $\{c_\lambda\}$ in an interesting way. One use of this latter result is as an informal quantitative check of a fitted moving average (or ARMA) model, by comparing the sample path behavior predicted by the model with the observed sample paths.

Citation

Download Citation

Holger Rootzen. "Extreme Value Theory for Moving Average Processes." Ann. Probab. 14 (2) 612 - 652, April, 1986. https://doi.org/10.1214/aop/1176992534

Information

Published: April, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0604.60019
MathSciNet: MR832027
Digital Object Identifier: 10.1214/aop/1176992534

Subjects:
Primary: 60F05
Secondary: 60G17

Keywords: ARMA processes , distributions of weighted sums , Extreme values , moving averages , properties , sample path

Rights: Copyright © 1986 Institute of Mathematical Statistics

JOURNAL ARTICLE
41 PAGES


SHARE
Vol.14 • No. 2 • April, 1986
Back to Top