Limit theory for the empirical extremogram of random fields

Regularly varying stochastic processes are able to model extremal dependence between process values at locations in random fields. We investigate the empirical extremogram as an estimator of dependence in the extremes. We provide conditions to ensure asymptotic normality of the empirical extremogram centred by a pre-asymptotic version. The proof relies on a CLT for exceedance variables. For max-stable processes with Fréchet margins we provide conditions such that the empirical extremogram centred by its true version is asymptotically normal. The results of this paper apply to a variety of spatial and space–time processes, and to time series models. We apply our results to max-moving average processes and Brown–Resnick processes.