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 depends on the local times of the differences between the logarithms of the underlying spectral processes.
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References
Barndorff-Nielsen, O.E., Corcuera, J.M., Podolskij, M.: Power variation for Gaussian processes with stationary increments. Stoch. Process. Appl. 119(6), 1845–1865 (2009)
Brown, B.M., Resnick, S.I.: Extreme values of independent stochastic processes. J. Appl. Probab. 14, 732–739 (1977)
de Haan, L., Ferreira, A.: Extreme Value Theory: An Introduction. Springer (2006)
de Haan, L., Lin, T.: On convergence towards an extreme value distribution in c[0,1]. Ann. Probab. 29, 467–483 (2001)
Dombry, C., Eyi-Minko, F.: Regular conditional distributions of continuous max-infinitely divisible random fields. Electron. J. Probab. 18(7), 1–21 (2013)
Einmahl, J.H.J., de Haan, L., Zhou, C.: Statistics of heteroscedastic extremes. Journal of the Royal Statistical Society, Series B 78, 31–51 (2016)
Ferreira, A., de Haan, L., Zhou, C.: Exceedance probability of the integral of a stochastic process. J. Multivar. Anal. 105, 241–257 (2012)
Giné, E., Hahn, M., Vatan, P.: Max-infinitely divisible and max-stable sample continuous processes. Probab. Theory Relat. Fields 87, 139–165 (1990)
Jacod, J.: Rates of convergence to the local time of a diffusion. Annales de l’Institut Henry Poincaré, Probab. Statist. 34, 505–544 (1998)
Jacod, J., Protter, P.: Discretization of Processes. Stochastic modelling and applied probability. Springer (2012)
Jacod, J., Rosenbaum, M.: Quarticity and other functionals of volatility: efficient estimation. Ann. Stat. 43, 1462–1484 (2013)
Jaramillo, A., Nourdin, I., Peccati, G.: Approximation of Fractional local times: Zero energy and derivatives. arXiv:1903.08683 (2019)
Kabluchko, Z., Schlather, M., de Haan, L.: Stationary max-stable fields associated to negative definite functions. Ann. Probab. 37, 2042–2065 (2009)
Li, J., Liu, Y., Xiu, D.: Efficient estimation of integrated volatility functionals via multiscale jackknife. Ann. Stat. 47, 156–176 (2019)
Rényi, A.: On stable sequences of events. Sankhya Ser. A 25, 293–302 (1963)
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer (1999)
Rosenbaum, M., Podolskij, M.: Asymptotic behavior of local times related statistics for fractional Brownian motion. J. Financ. Econ. 16(2018), 588–598 (2018)
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Robert, C.Y. Power variations for a class of Brown-Resnick processes. Extremes 23, 215–244 (2020). https://doi.org/10.1007/s10687-020-00373-4
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DOI: https://doi.org/10.1007/s10687-020-00373-4