The extremogram: A correlogram for extreme events

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 auto-correlation function, the extremogram, which depends only on the extreme values in the sequence. We also propose a natural estimation for the extremogram and study its asymptotic properties under alpha-mixing. We show asymptotic normality, calculate the extremogram for various examples and consider spectral analysis related to the extremogram.