Mandelbrot's 1/f fractional renewal models of 1963-67: The non-ergodic missing link between change points and long range dependence

TL;DR

This paper explores Mandelbrot's early work on 1/f noise and fractional renewal models from 1963-67, highlighting their non-ergodic nature and potential impact on understanding complexity, change points, and long-range dependence.

Contribution

It uncovers and discusses Mandelbrot's lesser-known early models, emphasizing their non-ergodic fractional renewal properties and their distinction from traditional LRD models.

Findings

Mandelbrot's 1963-67 work introduced non-ergodic fractional renewal models.

These models differ from stationary LRD models like fractional Gaussian noise.

The work influences the understanding of 1/f noise, change points, and the Hurst effect.

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, I present preliminary results of my research into the history of Mandelbrot's very little known work in that area from 1963-67. 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.