## Bernoulli

• Bernoulli
• Volume 15, Number 4 (2009), 977-1009.

### The extremogram: A correlogram for extreme events

#### Abstract

We consider a strictly stationary sequence of random vectors whose finite-dimensional distributions are jointly regularly varying with some positive index. This class of processes includes, among others, ARMA processes with regularly varying noise, GARCH processes with normally or Student-distributed noise and stochastic volatility models with regularly varying multiplicative noise. We define an analog of the autocorrelation function, the extremogram, which depends only on the extreme values in the sequence. We also propose a natural estimator for the extremogram and study its asymptotic properties under $α$-mixing. We show asymptotic normality, calculate the extremogram for various examples and consider spectral analysis related to the extremogram.

#### Article information

Source
Bernoulli, Volume 15, Number 4 (2009), 977-1009.

Dates
First available in Project Euclid: 8 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.bj/1262962223

Digital Object Identifier
doi:10.3150/09-BEJ213

Mathematical Reviews number (MathSciNet)
MR2597580

Zentralblatt MATH identifier
1200.62104

#### Citation

Davis, Richard A.; Mikosch, Thomas. The extremogram: A correlogram for extreme events. Bernoulli 15 (2009), no. 4, 977--1009. doi:10.3150/09-BEJ213. https://projecteuclid.org/euclid.bj/1262962223

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