Convergence of the Structure Function of a Multifractal Random Walk in a Mixed Asymptotic Setting |
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Authors: | Laurent Duvernet |
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Affiliation: | 1. Université Paris-Est Marne-la-Vallée and école Polytechnique , Marne-la-Vallée, France duvernet@cmap.polytechnique.fr |
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Abstract: | Some asymptotic properties of a Brownian motion in multifractal time, also called multifractal random walk, are established. We show the almost sure and L 1 convergence of its structure function. This is an issue directly connected to the scale invariance and multifractal property of the sample paths. We place ourselves in a mixed asymptotic setting where both the observation length and the sampling frequency may go together to infinity at different rates. The results we obtain are similar to the ones that were given by Ossiander and Waymire [19 Ossiander , M. , and Waymire , E.C. 2000 . Statistical estimation for multiplicative cascades . Ann. Stat. 28 : 1533 – 1560 .[Crossref], [Web of Science ®] , [Google Scholar]] and Bacry et al. [1 Bacry , E. , Gloter , A. , Hoffmann , M. , and Muzy , J.F. Multifractal analysis in a mixed asymptotic framework . Ann. Appl. Prob. (to appear) . [Google Scholar]] in the simpler framework of Mandelbrot cascades. |
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Keywords: | Multifractal analysis Multifractal random walks Scaling exponent |
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