Power variations for a class of Brown-Resnick processes

TL;DR

This paper develops the asymptotic theory for realized power variations of a specific class of Brown-Resnick max-stable processes, revealing bias depending on local times of spectral process differences.

Contribution

It introduces a biased central limit theorem for power variations of Brown-Resnick processes with continuous exponential martingale spectral processes.

Findings

Asymptotic normality with bias depending on local times

Infill asymptotic framework with high-frequency sampling

Theoretical characterization of power variations for max-stable processes

Abstract

We consider the class of simple Brown-Resnick max-stable processes whose spectral processes are continuous exponential martingales. We develop the asymptotic theory for the realized power variations of these max-stable processes, that is, sums of powers of absolute increments. We consider an infill asymptotic setting, where the sampling frequency converges to zero while the time span remains fixed. More specifically we obtain a biased central limit theorem whose bias depend on the local times of the differences between the logarithms of the underlying spectral processes.