On exact scaling log-Infinitely divisible cascades

TL;DR

This paper extends classical results on multiplicative cascades to exact scaling log-infinitely divisible cascades, providing conditions for non-degeneracy, moment finiteness, and analyzing tail behavior using a novel approach involving non-independent difference equations.

Contribution

It introduces a necessary and sufficient condition for non-degeneracy and moment finiteness of log-infinitely divisible cascades, extending prior results and applying Goldie's renewal theory to a new non-independent setting.

Findings

Provided a condition for non-degeneracy of limit measures.

Established criteria for finiteness of positive moments.

Analyzed the tail behavior of the total mass distribution.

Abstract

In this paper we extend some classical results valid for canonical multiplicative cascades to exact scaling log-infinitely divisible cascades. We complete previous results on non-degeneracy and moments of positive orders obtained by Barral and Mandelbrot, and Bacry and Muzy: we provide a necessary and sufficient condition for the non-degeneracy of the limit measures of these cascades, as well as for the finiteness of moments of positive orders of their total mass, extending Kahane's result for canonical cascades. Our main results are analogues to the results by Kahane and Guivarc'h regarding the asymptotic behavior of the right tail of the total mass. They rely on a new observation made about the cones used to define the log-infinitely divisible cascades; this observation provides a "non-independent" random difference equation satisfied by the total mass of the measures. The non-independent structure brings new difficulties to study the random difference equation, which we overcome thanks to Goldie's implicit renewal theory. We also discuss the finiteness of moments of negative orders, and some geometric properties of the support.