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


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.


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Holger Rootzen. "Extreme Value Theory for Moving Average Processes." Ann. Probab. 14 (2) 612 - 652, April, 1986.


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

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

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

Vol.14 • No. 2 • April, 1986
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