Multifractal processes: Definition, properties and new examples

Abstract

We investigate stochastic processes possessing scale invariance properties which we refer to as multifractal processes. The examples of such processes known so far do not go much beyond the original cascade construction of Mandelbrot. We provide a new definition of the multifractal process by generalizing the definition of the self-similar process. We establish general properties of these processes and show how existing examples fit into our setting. Finally, we define a new class of examples inspired by the idea of Lamperti transformation. Namely, for any pair of infinitely divisible distribution and a stationary process one can construct a multifractal process.

References in corpus (9)

• Log-infinitely divisible multifractal processes
• Multifractal stationary random measures and multifractal random walks with log-infinitely divisible scaling laws
• Multifractal analysis of the time series of daily means of wind speed in complex regions
• Gaussian multiplicative chaos and KPZ duality
• Lognormal scale invariant random measures
• Hydrodynamic turbulence and intermittent random fields
• On the Mellin transforms of the perpetuity and the remainder variables associated to a subordinator
• Mimicking self-similar processes
• Random cascade model in the limit of infinite integral scale as the exponential of a non-stationary $1/f$ noise. Application to volatility fluctuations in stock markets

Cited by in corpus (2)

• Description of turbulent dynamics in the interstellar medium: multifractal/microcanonical analysis I. Application to Herschel observations of the Musca filament
• Integer-valued multifractal processes