Toward a deeper understanding of a basic cascade

Abstract

The study of turbulence and chaos has led to crucial new concepts. In particular, B. Mandelbrot introduced and promoted (multi-)fractals. More recently, cascades have gained momentum. Not least due to technical difficulties, continuous stochastic models, such as the classic p model, have been preferred over discrete cascades. It is the aim of this contribution to introduce original concepts that shed new light on a variant of the latter paradigmatic setting 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 that is based on a new kind of sampling, named “power sampling.” Technical machinery can be kept simple; therefore, proofs are straightforward and formulas explicit. It is hoped that the proposed line of investigation may enhance understanding and simplify received multi-fractal analysis.

Keywords:

• Cascades
• p model
• multifractals
• sampling
• law of large numbers
• Mandelbrot

Acknowledgments

The author would like to thank two anonymous referees for very valuable suggestions that improved the paper considerably.

Dedication

This paper is dedicated to the memory of my teacher Klaus Janßen (1942–2024), Professor emeritus at Heinrich-Heine Universität Düsseldorf, Germany.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

Data Availability Statement: NA

Notes

1. See the remark after the proof of Theorem 2, p. 2

2. Pascal named his triangle “triangle arithmetique.” Thus, at least in French, it is straightforward to name the above multiplicative structure “triangle geometrique.” In English, we prefer ‘binary’ or ‘Bernoulli cascade’.

3. It may be noted that the “Weaver” is similar to the “baker” in dynamic system theory. In particular, in both cases, a locally defined transformation is closely related to global patterns. Theorem 9 connects the stochastic and the dynamic points of view explicitly.