Generalised cascades

In this manuscript we give thought to the aftermath on the stable probability density function when standard multiplicative cascades are generalised cascades based on the $q$-product of Borges that emerged in the context of non-extensive statistical mechanics.

💡 Research Summary

The manuscript investigates how the probability density function (PDF) of a stable distribution is altered when the classic multiplicative cascade is replaced by a generalized cascade built on Borges’ q‑product, an operation that originates from non‑extensive statistical mechanics. The authors begin by recalling the standard multiplicative cascade: a sequence of independent positive random variables (X_i) is multiplied stepwise, yielding a final variable (Y_N=\prod_{i=1}^{N}X_i). Under the usual logarithmic transformation, the sum (\ln Y_N=\sum_i\ln X_i) obeys the central limit theorem, leading to a log‑normal limit for large (N).

The core innovation is the introduction of the q‑product, \