## Bernoulli

• Bernoulli
• Volume 22, Number 3 (2016), 1671-1708.

### Functional limit theorems for generalized variations of the fractional Brownian sheet

#### Abstract

We prove functional central and non-central limit theorems for generalized variations of the anisotropic $d$-parameter fractional Brownian sheet (fBs) for any natural number $d$. Whether the central or the non-central limit theorem applies depends on the Hermite rank of the variation functional and on the smallest component of the Hurst parameter vector of the fBs. The limiting process in the former result is another fBs, independent of the original fBs, whereas the limit given by the latter result is an Hermite sheet, which is driven by the same white noise as the original fBs. As an application, we derive functional limit theorems for power variations of the fBs and discuss what is a proper way to interpolate them to ensure functional convergence.

#### Article information

Source
Bernoulli, Volume 22, Number 3 (2016), 1671-1708.

Dates
Revised: January 2015
First available in Project Euclid: 16 March 2016

https://projecteuclid.org/euclid.bj/1458132995

Digital Object Identifier
doi:10.3150/15-BEJ707

Mathematical Reviews number (MathSciNet)
MR3474829

Zentralblatt MATH identifier
1338.60099

#### Citation

Pakkanen, Mikko S.; Réveillac, Anthony. Functional limit theorems for generalized variations of the fractional Brownian sheet. Bernoulli 22 (2016), no. 3, 1671--1708. doi:10.3150/15-BEJ707. https://projecteuclid.org/euclid.bj/1458132995

#### References

• [1] Ayache, A., Leger, S. and Pontier, M. (2002). Drap brownien fractionnaire. Potential Anal. 17 31–43.
• [2] Bardet, J.-M. and Surgailis, D. (2013). Moment bounds and central limit theorems for Gaussian subordinated arrays. J. Multivariate Anal. 114 457–473.
• [3] Bardina, X. and Florit, C. (2005). Approximation in law to the $d$-parameter fractional Brownian sheet based on the functional invariance principle. Rev. Mat. Iberoam. 21 1037–1052.
• [4] Barndorff-Nielsen, O.E., Corcuera, J.M. and Podolskij, M. (2009). Power variation for Gaussian processes with stationary increments. Stochastic Process. Appl. 119 1845–1865.
• [5] Barndorff-Nielsen, O.E. and Graversen, S.E. (2011). Volatility determination in an ambit process setting. J. Appl. Probab. 48A 263–275.
• [6] Bickel, P.J. and Wichura, M.J. (1971). Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Stat. 42 1656–1670.
• [7] Breuer, P. and Major, P. (1983). Central limit theorems for nonlinear functionals of Gaussian fields. J. Multivariate Anal. 13 425–441.
• [8] Clarke De la Cerda, J. and Tudor, C.A. (2014). Wiener integrals with respect to the Hermite random field and applications to the wave equation. Collect. Math. 65 341–356.
• [9] Dobrushin, R.L. and Major, P. (1979). Non-central limit theorems for nonlinear functionals of Gaussian fields. Z. Wahrsch. Verw. Gebiete 50 27–52.
• [10] Giraitis, L. and Surgailis, D. (1985). CLT and other limit theorems for functionals of Gaussian processes. Z. Wahrsch. Verw. Gebiete 70 191–212.
• [11] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Probability and Its Applications (New York). New York: Springer.
• [12] Maejima, M. and Tudor, C.A. (2013). On the distribution of the Rosenblatt process. Statist. Probab. Lett. 83 1490–1495.
• [13] Mandelbrot, B.B. and Van Ness, J.W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 422–437.
• [14] Norros, I., Valkeila, E. and Virtamo, J. (1999). An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli 5 571–587.
• [15] Nourdin, I., Nualart, D. and Tudor, C.A. (2010). Central and non-central limit theorems for weighted power variations of fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 46 1055–1079.
• [16] Nourdin, I. and Peccati, G. (2009). Stein’s method on Wiener chaos. Probab. Theory Related Fields 145 75–118.
• [17] Nourdin, I., Peccati, G. and Podolskij, M. (2011). Quantitative Breuer–Major theorems. Stochastic Process. Appl. 121 793–812.
• [18] Nourdin, I., Peccati, G. and Réveillac, A. (2010). Multivariate normal approximation using Stein’s method and Malliavin calculus. Ann. Inst. Henri Poincaré Probab. Stat. 46 45–58.
• [19] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Probability and Its Applications (New York). Berlin: Springer.
• [20] Nualart, D. and Peccati, G. (2005). Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 177–193.
• [21] Pakkanen, M.S. (2014). Limit theorems for power variations of ambit fields driven by white noise. Stochastic Process. Appl. 124 1942–1973.
• [22] Peccati, G. and Tudor, C.A. (2005). Gaussian limits for vector-valued multiple stochastic integrals. In Séminaire de Probabilités XXXVIII. Lecture Notes in Math. 1857 247–262. Berlin: Springer.
• [23] Rényi, A. (1963). On stable sequences of events. Sankhyā Ser. A 25 293–302.
• [24] Réveillac, A. (2009). Convergence of finite-dimensional laws of the weighted quadratic variations process for some fractional Brownian sheets. Stoch. Anal. Appl. 27 51–73.
• [25] Réveillac, A., Stauch, M. and Tudor, C.A. (2012). Hermite variations of the fractional Brownian sheet. Stoch. Dyn. 12 1150021.
• [26] Rosenblatt, M. (1961). Independence and dependence. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II 431–443. Berkeley, Calif.: Univ. California Press.
• [27] Rosenblatt, M. (1981). Limit theorems for Fourier transforms of functionals of Gaussian sequences. Z. Wahrsch. Verw. Gebiete 55 123–132.
• [28] Taqqu, M.S. (1974/75). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 287–302.
• [29] Taqqu, M.S. (1977). Law of the iterated logarithm for sums of non-linear functions of Gaussian variables that exhibit a long range dependence. Z. Wahrsch. Verw. Gebiete 40 203–238.
• [30] Taqqu, M.S. (1978). A representation for self-similar processes. Stochastic Process. Appl. 7 55–64.
• [31] Taqqu, M.S. (1979). Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrsch. Verw. Gebiete 50 53–83.
• [32] Tudor, C.A. (2008). Analysis of the Rosenblatt process. ESAIM Probab. Stat. 12 230–257.
• [33] Veillette, M.S. and Taqqu, M.S. (2013). Properties and numerical evaluation of the Rosenblatt distribution. Bernoulli 19 982–1005.