Abstract
The problem of 1/f noise was identified by physicists about a century ago, while the puzzle posed by Hurst’s eponymous effect, originally identified by statisticians, hydrologists and time series analysts, is over 60 years old. Because these communities so often frame the problems in Fourier spectral language, the most famous solutions have tended to be the stationary ergodic 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 processes (FRP), I present the first results of my research into the history of Mandelbrot’s very little known work on the FRP in 1963–67. I discuss the differences between the Hurst effect, 1/f noise and LRD, concepts which are often treated as equivalent, and finally speculate about how the lack of awareness of his FRP papers in the physics and statistics communities may have affected the development of complexity science.
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Acknowledgements
I would like to thank Rebecca Killick for inviting me to talk at ITISE 2016, and helpful comments on an earlier version from Eli Barkai. I also gratefully acknowledge many valuable discussions about the history of LRD and weak ergodicity breaking with Nick Moloney, Christian Franzke, Ralf Metzler, Holger Kantz, Igor Sokolov, Rainer Klages, Tim Graves, Bobby Gramacy, Andrey Cherstvy, Aljaz Godec, Sandra Chapman, Thordis Thorarinsdottir, Kristoffer Rypdal, Martin Rypdal, Bogdan Hnat, Daniela Froemberg, and Igor Goychuk among many others. I acknowledge travel support from KLIMAFORSK project number 229754 and the London Mathematical Laboratory, a senior visiting fellowship from the Max Planck Society in Dresden, and Office of Naval Research NICOP grant NICOP-N62909-15-1-N143 at Warwick and Potsdam.
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Wynn Watkins, N. (2017). Mandelbrot’s 1/f Fractional Renewal Models of 1963–67: The Non-ergodic Missing Link Between Change Points and Long Range Dependence. In: Rojas, I., Pomares, H., Valenzuela, O. (eds) Advances in Time Series Analysis and Forecasting. ITISE 2016. Contributions to Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-55789-2_14
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