Weak convergence to Rosenblatt sheet

Abstract

We study the problem of the approximation in law of the Rosenblatt sheet. We prove the convergence in law of two families of process to the Rosenblatt sheet: the first one is constructed from a Poisson process in the plane and the second one is based on random walks.

Keywords

Rosenblatt sheet / Poisson process / random walks / weak convergence

References

[1] Abry P, Pipiras V. Wavelet-based synthesis of the Rosenblatt process. Signal Process, 2006, 86: 2326-2339

[2] Albin J M P. On extremal theory for self similar processes. Ann Probab, 1998, 26: 743-793

[3] Bardina X, Florit C. Approximation in law to the d-parameter fractional Brownian sheet based on the functional invariance principle. Rev Mat Iberoam, 2005, 21: 1037-1052

[4] Bardina X, Jolis M. Weak approximation of the Brownian sheet from the Poisson process in the plane. Bernoulli, 2000, 6(4): 653-665

[5] Bardina X, Jolis M, Tudor C A. Weak convergence to the fractional Brownian sheet and other two-parameter Gaussian processes. Statist Probab Lett, 2003, 65: 317-329

[6] Biagini F, Hu Y, Øksendal B, Zhang T. Stochastic Calculus for fBm and Applications. New York: Springer-Verlag, 2008

[7] Bickel P, Wichura M. Convergence criteria for multiparamenter stochastic process and some applications. Ann Math Statist, 1971, 42: 1656-1670

[8] Cairoli R, Walsh J. Stochastic integrals in the plane. Acta Math, 1975, 134: 111-183

[9] Chen C, Sun L, Yan L. An approximation to the Rosenblatt process using martingale differences. Statist Probab Lett, 2012, 82: 748-757

[10] Chronopoulou A, Tudor C, Viens F. Variations and Hurst index estimation for a Rosenblatt process using longer filters. Electron J Stat, 2009, 3: 1393-1435

[11] Dai H. Convergence in law to operator fractional Brownian motions. J Theoret Probab, 2013, 26: 676-696

[12] Dobrushin R L, Major P. Non-central limit theorems for non-linear functionals of Gaussian fields. Z Wahrschein-lichkeitstheor Verwandte Geb, 1979, 50: 27-52

[13] Hall P, Hardle W, Kleinow T, Schmidt P. Semiparametric bootstrap approach to hypothesis tests and confidence intervals for the Hurst coefficient. Stat Inference Stoch Process, 2000, 3: 263-276

[14] Leonenko N N, Ahn V V. Rate of convergence to the Rosenblatt distribution for additive functionals of stochastic processes with long-range dependence. J Appl Math Stoch Anal, 2001, 14: 27-46

[15] Leonenko N N, Woyczynski W. Scaling limits of solutions of the heat equation for singular non-Gaussian data. J Stat Phys, 1998, 91: 423-438

[16] Li Y, Dai H. Approximations of fractional Brownian motion. Bernoulli, 2011, 17: 1195-1216

[17] Maejima M, Tudor C A. Wiener integrals with respect to the Hermite process and a non central limit theorem. Stoch Anal Appl, 2007, 25: 1043-1056

[18] Maejima M, Tudor C A. Selfsimilar processes with stationary increments in the second Wiener chaos. Probab Math Statist, 2012, 32: 167-186

[19] Maejima M, Tudor C A. On the distribution of the Rosenblatt process. Statist Probab Lett, 2013, 83: 1490-1495

[20] Mishura Y. Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Math, Vol 1929. Berlin: Springer, 2008

[21] Pipiras V, Taqqu M S. Regularization and integral representations of Hermite processes. Statist Probab Lett, 2010, 80: 2014-2023

[22] Shen G, Ren Y. Neutral stochastic partial differential equations with delay driven by Rosenblatt process in a Hilbert space. J Korean Statist Soc, 2014, 44: 123-133

[23] Shieh N-R, Xiao Y. Hausdorff and packing dimensions of the images of random fields. Bernoulli, 2010, 16: 926-952

[24] Sottinen T. Fractional Brownian motion, random walks and binary market models. Finance Stoch, 2001, 5: 343-355

[25] Stroock D. Topics in Stochastic Differential Equations. Berlin: Springer-Verlag, 1982

[26] Taqqu M. Weak convergence to the fractional Brownian motion and to the Rosenblatt process. Z Wahrschein-lichkeitstheor Verwandte Geb, 1975, 31: 287-302

[27] Taqqu M. Convergence of integrated processes of arbitrary Hermite rank. Zwahrscheinlichkeitstheor Verwandte Geb, 1979, 50: 53-83

[28] Torres S, Tudor C. Donsker type theorem for the Rosenblatt process and a binary market model. Stoch Anal Appl, 2009, 27: 555-573

[29] Tudor C. Weak convergence to the fractional Brownian sheet in Besov spaces. Bull Braz Math Soc, 2003, 34: 389-400

[30] Tudor C. Analysis of the Rosenblatt process. ESAIM Probab Stat, 2008, 12: 230-257

[31] Tudor C. Analysis of Variations for Self-similar Processes. Berlin: Springer, 2013

[32] Tudor C, Viens F. Variations and estimators for the selfsimilarity order through Malliavin calculus. Ann Probab, 2009, 37: 2093-2134

[33] Wang Z, Yan L, Yu X. Weak approximation of the fractional Brownian sheet from random walks. Electron Commun Probab, 2013, 18: 1-13

[34] Wang Z, Yan L, Yu X. Weak approximation of the fractional Brownian sheet using martingale differences. Statist Probab Lett, 2014, 92: 72-78

[35] Wu W B. Unit root testing for functionals of linear processes. Econom Theory, 2005, 22: 1-14

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Higher Education Press and Springer-Verlag Berlin Heidelberg