Asymptotic behavior of the quadratic variation of the sum of two Hermite processes of consecutive orders |
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Authors: | M Clausel F Roueff MS Taqqu C Tudor |
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Institution: | 1. Laboratoire Jean Kuntzmann, Université de Grenoble-Alpes, CNRS, F38041 Grenoble Cedex 9, France;2. Institut Mines-Telecom, Telecom ParisTech, CNRS LTCI, 46 rue Barrault, 75634 Paris Cedex 13, France;3. Department of Mathematics and Statistics, Boston University, Boston, MA 02215, USA;4. Laboratoire Paul Painlevé, UMR 8524 du CNRS, Université Lille 1, 59655 Villeneuve d’Ascq, France;5. Department of Mathematics, Academy of Economical Studies, Bucharest, Romania |
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Abstract: | 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. |
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Keywords: | primary 60G18 60G22 secondary 60H05 |
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