Limit theorems for power variations of ambit fields driven by white noise

Abstract

We study the asymptotics of lattice power variations of two-parameter ambit fields driven by white noise. Our first result is a law of large numbers for power variations. Under a constraint on the memory of the ambit field, normalized power variations converge to certain integral functionals of the volatility field associated to the ambit field, when the lattice spacing tends to zero. This result holds also for thinned power variations that are computed by only including increments that are separated by gaps with a particular asymptotic behavior. Our second result is a stable central limit theorem for thinned power variations.

31 pages, 2 figures, v4: several improvements, including new remarks on the underlying ambit field and on a more general observation scheme

References in corpus (4)

• Central limit theorems for sequences of multiple stochastic integrals
• Multipower variation for Brownian semistationary processes
• Asymptotic theory for Brownian semi-stationary processes with application to turbulence
• Assessing Relative Volatility/Intermittency/Energy Dissipation

Cited by in corpus (7)

• High-frequency analysis of parabolic stochastic PDEs
• Assessing Relative Volatility/Intermittency/Energy Dissipation
• Functional limit theorems for generalized variations of the fractional Brownian sheet
• Modelling spatial heteroskedasticity by volatility modulated moving averages
• Power variations for fractional type infinitely divisible random fields
• Local scaling limits of Lévy driven fractional random fields
• Stationary and multi-self-similar random fields with stochastic volatility