Asymptotic behavior of the quadratic variation of the sum of two Hermite processes of consecutive orders

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:124

Article

Clausel, M.¹  Roueff, F.²  Taqqu, M. S.³  Tudor, C.^4,5 

[1] Univ Grenoble Alpes, CNRS, Lab Jean Kuntzmann, F-38041 Grenoble 9, France

[2] Telecom ParisTech, CNRS LTC, Inst Mines Telecom, F-75634 Paris 13, France

[3] Boston Univ, Dept Math & Stat, Boston, MA 02215 USA

[4] Univ Lille 1, CNRS, Lab Paul Painleve, UMR 8524, F-59655 Villeneuve Dascq, France

[5] Acad Econ Studies, Dept Math, Bucharest, Romania

关键词: Hermite processes;    Quadratic variation;    Covariation;    Wiener chaos;    Self-similar processes;    Long-range dependence;   

DOI  :  10.1016/j.spa.2014.02.013

来源: Elsevier

【 摘 要 】

Hermite processes are self-similar processes with stationary increments which appear as limits of normalized sums of random variables with long range dependence. The Hermite process of order 1 is fractional Brownian motion and the Hermite process of order 2 is the Rosenblatt process. We consider here the sum of two Hermite processes of orders q >= 1 and q + 1 and of different Hurst parameters. We then study its quadratic variations at different scales. This is akin to a wavelet decomposition. We study both the cases where the Hermite processes are dependent and where they are independent. In the dependent case, we show that the quadratic variation, suitably normalized, converges either to a normal or to a Rosenblatt distribution, whatever the order of the original Hermite processes. (C) 2014 Elsevier B.V. All rights reserved.

【 授权许可】

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