02 Sep 2019

02 Sep 2019

Review status: this preprint is currently under review for the journal NPG.

In-depth analysis of a discrete p model

Uwe Saint-Mont Uwe Saint-Mont
  • Nordhausen University of Applied Sciences

Abstract. Towards the end of the last century, B. Mandelbrot saw the importance, revealed the beauty, and robustly promoted (multi)fractals. Multiplicative cascades are closely related and provide simple models for the study of turbulence and chaos.

For pedagogical reasons, but also due to technical difficulties, continuous stochastic models have been favoured over discrete cascades. Particularly important are the α, the β and the p model (Lovejoy and Scherzter 2013, Chapter 3; de Wijs (1951, 1953). It is the aim of this contribution to introduce original concepts that shed new light on the latter paradigmatic cascade and allow key features to be derived in a rather elementary fashion.

To this end, we introduce and study a discrete version of the p model which is based on a new kind of sampling. Technical machinery can be kept simple, therefore formulas are explicit, proofs extend standard arguments, and potential extensions are numerous. Thus the proposed line of investigation may enrich and simplify received multifractal analyses.

Uwe Saint-Mont

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Uwe Saint-Mont

Uwe Saint-Mont


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Short summary
Every student of probability knows Pascal's triangle. Mandelbrot's fractals are also rather well known; more recently, cascades have come into focus. This article starts with 'local' cascades and shows that they are equivalent to a new 'global' operation (weaving). Thus, one obtains a multiplicative pattern that lies at the heart of a new discrete 'p model.' Since the technical machinery is simple, corresponding distributions can be analysed in detail, extending received results considerably.