On the continuing relevance of Mandelbrot’s non-ergodic fractional renewal models of 1963 to 1967

Abstract

Abstract. 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.

1/f noise and anomalous scaling in Lévy noise-driven on–off intermittency

“…Such behaviour, namely the existence of a wide range in log(f ), at small f, for which the PSD S (f ) is of power-law form with exponent smaller than 0 and greater than −2, is generically called 1/f noise, also known as Flicker noise, or pink noise. It has been observed in a wide variety of systems, ranging from voltage and current fluctuations in vacuum tubes and transistors, where this behaviour was first recognised [44][45][46], to blinking dots [47,48], to astrophysical magnetic fields [49] and biological systems [50], climate [51], turbulent flows [52][53][54], reversing flows [55][56][57][58], traffic [59], as well as music and speech [60,61], to name a few, and is also found in fractional renewal models [62]. In addition, 1/f noise has also been observed for Lévy flights in inhomogeneous environments [63,64], but these studies did not consider any bifurcation points.…”

Meditation-Induced Coherence and Crucial Events

“…One of the most difficult issues confronting complexity science is the origin of 1/ f -noise (Watkins, 2016 , 2017 ). Is it an ergodic or a non-ergodic process?…”

“…Is it an ergodic or a non-ergodic process? Watkins ( 2017 ) has recently pointed out that Mandelbrot, very well known for his generalization of ordinary diffusion, called Fractional Brownian Motion (FBM) (Mandelbrot and Van Ness, 1968 ), which has a physical origin compatible with stationarity and ergodicity, is also the author of papers (Mandelbrot, 1965 , 1967 ) opening a bridge between the stationary and non-stationary condition. Actually, the revisitation of the work done by Mandelbrot in the 1963-1967 period of time leads to the creation of a connection with the Continuous Time Random Walk (CTRW) (Montroll and Weiss, 1965 ; Shlesinger, 2017 ).…”

“…According to Watkins ( 2016 , 2017 ) the non-ergodic fractional renewal model of Mandelbrot ( 1965 , 1967 ) yields the inverse power law (IPL) spectrum as a function of the frequency f :…”

“…Consequently, the system evolves toward the occurrence of the next renewal event as if the occurrence of the earlier crucial event made the system brand new. The renewal property facilitates resolving the 1/ f − paradox (Watkins, 2016 , 2017 ). This paradox has been brilliantly discussed by Niemann et al ( 2013 ) who noticed that for γ > 1(μ < 2) the spectrum S ( f ) is not integrable, thereby violating the integrability condition on which the Wiener-Khinchine theorem is based.…”