## The Annals of Probability

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

### Extreme Value Theory for Moving Average Processes

#### 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