The Annals of Probability

Extreme Value Theory for Moving Average Processes

Holger Rootzen

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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.

Article information

Source
Ann. Probab., Volume 14, Number 2 (1986), 612-652.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176992534

Digital Object Identifier
doi:10.1214/aop/1176992534

Mathematical Reviews number (MathSciNet)
MR832027

Zentralblatt MATH identifier
0604.60019

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60G17: Sample path properties

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

Citation

Rootzen, Holger. Extreme Value Theory for Moving Average Processes. Ann. Probab. 14 (1986), no. 2, 612--652. doi:10.1214/aop/1176992534. https://projecteuclid.org/euclid.aop/1176992534


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