On the continuing relevance of Mandelbrot’s non-ergodic fractional renewal models of 1963 to 1967★
Centre for the Analysis of Time Series, London School of Economics and Political Science,
2 Centre for Fusion, Space and Astrophysics, University of Warwick, Coventry, UK
3 Faculty of Science, Technology, Engineering and Mathematics, Open University, Milton Keynes, UK
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Received in final form: 25 September 2017
Published online: 11 December 2017
The problem of “1∕f” noise has been with us for about a century. Because it is so often framed in Fourier spectral language, the most famous solutions have tended to be the stationary long range dependent (LRD) models such as Mandelbrot’s fractional Gaussian noise. In view of the increasing importance to physics of non-ergodic fractional renewal models, and their links to the CTRW, I present preliminary results of my research into the history of Mandelbrot’s very little known work in that area from 1963 to 1967. I speculate about how the lack of awareness of this work in the physics and statistics communities may have affected the development of complexity science, and I discuss the differences between the Hurst effect, “1∕f” noise and LRD, concepts which are often treated as equivalent.
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