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Part of the book series: Probability and Its Applications ((PIA))

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Abstract

Chapter 3 is consecrated to the study of the Hermite processes. The Hermite processes are self-similar processes with stationary increments. They appear as limits in the so called Non-Central Limit Theorem. This class includes the fractional Brownian motion but all the other processes in the class of Hermite processes are non-Gaussian. Another interesting example in this class is the Rosenblatt process, which is discussed is details, together with its variants, in this part of the monograph.

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References

  1. J.M.P. Albin, A note on the Rosenblatt distributions. Stat. Probab. Lett. 40(1), 83–91 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. J.M.P. Albin, On extremal theory for self similar processes. Ann. Probab. 26(2), 743–793 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. E. Alos, D. Nualart, Stochastic integration with respect to the fractional Brownian motion. Stoch. Stoch. Rep. 75, 129–152 (2003)

    Article  MathSciNet  Google Scholar 

  4. T. Androshuk, Y. Mishura, Mixed Brownian-fractional Brownian model: absence of arbitrage and related topics. Stoch. Int. J. Probab. Stoch. Process. 78, 281–300 (2006)

    Article  Google Scholar 

  5. K. Bertin, S. Torres, C.A. Tudor, Maximum likelihood estimators and random walks in long-memory models. Stat. J. Theor. Appl. Stat. 45(4), 361–374 (2011)

    MathSciNet  MATH  Google Scholar 

  6. S. Bonaccorsi, C.A. Tudor, Dissipative stochastic evolution equations driven by general Gaussian and non-Gaussian noise. J. Dyn. Differ. Equ. 23(4), 791–816 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. C. Chen, L. Sun, L. Yan, An approximation to the Rosenblatt process using martingale differences. Stat. Probab. Lett. 82(4), 748–757 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. P. Cheridito, H. Kawaguchi, M. Maejima, Fractional Ornstein-Uhlenbeck processes. Electron. J. Probab. 8, 1–14 (2003)

    Article  MathSciNet  Google Scholar 

  9. R.L. Dobrushin, P. Major, Non-central limit theorems for non-linear functionals of Gaussian fields. Z. Wahrscheinlichkeitstheor. Verw. Geb. 50, 27–52 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Embrechts, M. Maejima, Selfsimilar Processes (Princeton University Press, Princeton, 2002)

    MATH  Google Scholar 

  11. R. Fox, M.S. Taqqu, Multiple stochastic integrals with dependent integrators. J. Multivar. Anal. 21, 105–127 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Garzón, S. Torres, C.A. Tudor, A strong convergence to the Rosenblatt process. J. Math. Anal. Appl. 391(2), 630–647 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Y.Z. Hu, P.A. Meyer, Sur les intégrales multiples de Stratonovich, in Séminaire de Probabilités XXII. Lecture Notes in Mathematics (1988), pp. 72–81

    Chapter  Google Scholar 

  14. A.J. Lawrance, N.T. Kottegoda, Stochastic modelling of riverflow time series. J. R. Stat. Soc. Ser. A 140(1), 1–47 (1977)

    Article  Google Scholar 

  15. M. Maejima, C.A. Tudor, Wiener integrals and a Non-central limit theorem for Hermite processes. Stoch. Anal. Appl. 25(5), 1043–1056 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Maejima, C.A. Tudor, Selfsimilar processes with stationary increments in the second Wiener chaos. Probab. Math. Stat. 32(1), 167–186 (2012)

    MathSciNet  MATH  Google Scholar 

  17. M. Maejima, C.A. Tudor, On the distribution of the Rosenblatt process. Preprint (2012)

    Google Scholar 

  18. T. Mori, H. Oodaira, The law of the iterated logarithm for self-similar processes represented by multiple Wiener integrals. Probab. Theory Relat. Fields 71, 367–391 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  19. I. Nourdin, G. Peccati, Cumulants on Wiener space. J. Funct. Anal. 258, 3775–3791 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. I. Nourdin, D. Nualart, C.A. Tudor, Central and non-central limit theorems for weighted power variations of fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 46(4), 1055–1079 (2007)

    Article  MathSciNet  Google Scholar 

  21. D. Nualart, Malliavin Calculus and Related Topics, 2nd edn. (Springer, New York, 2006)

    MATH  Google Scholar 

  22. V. Pipiras, Wavelet type expansion of the Rosenblatt process. J. Fourier Anal. Appl. 10(6), 599–634 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. V. Pipiras, M. Taqqu, Integration questions related to the fractional Brownian motion. Probab. Theory Relat. Fields 118(2), 251–281 (2001)

    Article  MathSciNet  Google Scholar 

  24. V. Pipiras, M. Taqqu, Regularization and integral representations of Hermite processes. Stat. Probab. Lett. 80, 2014–2023 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. M. Rosenblatt, Independence and dependence, in Proc. 4th Berkeley Symposium on Math. Stat., vol. II (1961), pp. 431–443

    Google Scholar 

  26. J.L. Sole, F. Utzet, Stratonovich integral and trace. Stoch. Stoch. Rep. 29(2), 203–220 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  27. M. Taqqu, Weak convergence to the fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitstheor. Verw. Geb. 31, 287–302 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Taqqu, Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrscheinlichkeitstheor. Verw. Geb. 50, 53–83 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  29. M. Taqqu, A representation for self-similar processes. Stoch. Process. Appl. 7, 55–64 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  30. S. Torres, C.A. Tudor, Donsker type theorem for the Rosenblatt process and a binary market model. Stoch. Anal. Appl. 27, 555–573 (2008)

    Article  MathSciNet  Google Scholar 

  31. C.A. Tudor, Analysis of the Rosenblatt Process. ESAIM Probability and Statistics, vol. 12, (2008), pp. 230–257

    Google Scholar 

  32. M.S. Veillette, M.S. Taqqu, Properties and numerical evaluation of the Rosenblatt distribution. Bernoulli 19(3) 982–1005 (2013)

    Article  MathSciNet  Google Scholar 

  33. W. Wilinger, M. Taquu, A. Erramilli, A bibliographical guide to self-similar traffic and performance modeling for modern high-speed networks, in Stochastic Networks: Theory and Applications (Clarendon Press, Oxford, 1996), pp. 339–366

    Google Scholar 

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Tudor, C.A. (2013). Non-Gaussian Self-similar Processes. In: Analysis of Variations for Self-similar Processes. Probability and Its Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-00936-0_3

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