Abstract
Let ρ be a real-valued function on [0, T], and let LSI(ρ) be a class of Gaussian processes over time interval [0, T], which need not have stationary increments but their incremental variance σ(s, t) is close to the values ρ(|t − s|) as t → s uniformly in s ∈ (0, T]. For a Gaussian processesGfrom LSI(ρ), we consider a power variation V n corresponding to a regular partition π n of [0, T] and weighted by values of ρ(·). Under suitable hypotheses on G, we prove that a central limit theorem holds for V n as the mesh of π n approaches zero. The proof is based on a general central limit theorem for random variables that admit a Wiener chaos representation. The present result extends the central limit theorem for a power variation of a class of Gaussian processes with stationary increments and for bifractional and subfractional Gaussian processes.
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References
S. Aazizi and K. Es-Sebaiy, Berry–Esséen bounds and almost sure CLT for the quadratic variation of the bifractional Brownian motion, 2012, arXiv:1203.2786v3.
O.E. Barndorff-Nielsen, J.M. Corcuera, and M. Podolskij, Power variation for Gaussian processes with stationary increments, Stoch. Process. Appl., 119(6):1845–1865, 2009.
G. Baxter, A strong limit theorem for Gaussian processes, Proc. Am. Math. Soc., 7:522–527, 1956.
T. Bojdecki, L.G. Gorostiza, and A. Talarczyk, Sub-fractional Brownian motion and its relation to occupation time, Stat. Probab. Lett., 69:405–419, 2004.
E.G. Gladyshev, A new limit theorem for stochastic processes with Gaussian increments, Theory Probab. Appl., 6:52–61, 1961.
X. Guyon and J. Leon, Convergence en loi des H-variations d’un processus gaussien stationnaire sur R, Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. B, 25(3):265–282, 1989 (in French).
C. Houdré and J. Villa, An example of infinite dimensional quasi-helix, in Stochastic Models: Seventh Symposium on Probability and Stochastic Processes (Mexico City, June 23–28, 2002), Contemp. Math., Vol. 366, Amer. Math. Soc., Providence, RI, 2003, pp. 195–201.
Y. Hu and D. Nualart, Renormalized self-intersection local time for fractional Brownian motion, Ann. Probab., 33:948–983, 2005.
P. Lévy, Le mouvement brownien plan, Am. J. Math., 62:487–550, 1940 (in French).
R. Norvaiša, Weighted power variation of integrals with respect to a Gaussian process, Bernoulli, 2014 (in press).
D. Nualart, The Malliavin Calculus and Related Topics, 2nd ed., Probability and Its Applications, Springer, New York, 2006.
C. Tudor, Berry–Esséen bounds and almost sure CLT for the quadratic variation of the sub-fractional Brownian motion, J. Math. Anal. Appl., 375(2):667–676, 2011.
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*This research was funded by a grant (No. MIP-053/2012) from the Research Council of Lithuania.
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Malukas, R., Norvaiša, R. A central limit theorem for a weighted power variation of a Gaussian process*. Lith Math J 54, 323–344 (2014). https://doi.org/10.1007/s10986-014-9246-8
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DOI: https://doi.org/10.1007/s10986-014-9246-8