Multifractal scaling of products of birth--death processes

TL;DR

This paper studies the multifractal scaling behavior of products of exponentiated birth-death processes with specific distributions, analyzing their properties and limiting behaviors.

Contribution

It introduces a framework for understanding the scaling and dependence of products of birth-death processes with various distributions, including explicit examples and limiting processes.

Findings

Derived conditions for mean, variance, and covariance functions.

Provided explicit examples with Poisson, Pascal, binomial, and hypergeometric distributions.

Established limiting processes and their multifractal properties.

Abstract

We investigate the scaling properties of products of the exponential of birth--death processes with certain given marginal discrete distributions and covariance structures. The conditions on the mean, variance and covariance functions of the resulting cumulative processes are interpreted in terms of the moment generating functions. We provide four illustrative examples of Poisson, Pascal, binomial and hypergeometric distributions. We establish the corresponding log-Poisson, log-Pascal, log-binomial and log-hypergeometric scenarios for the limiting processes, including their R\'{e}nyi functions and dependence properties.