Limit Theorems for Multifractal Products of Geometric Stationary Processes

TL;DR

This paper studies the multifractal properties of geometric Gaussian and Ornstein-Uhlenbeck processes with long-range dependence, providing conditions for convergence and analyzing their multifractal spectra.

Contribution

It offers new general conditions for $L_q$ convergence and characterizes the multifractal behavior of various geometric processes driven by Lévy motion.

Findings

Established conditions for $L_q$ convergence of multifractal processes.

Analyzed the multifractal spectra and Rényi functions of the processes.

Identified scenarios like log-normal and stable distributions for limiting processes.

Abstract

We investigate the properties of multifractal products of geometric Gaussian processes with possible long-range dependence and geometric Ornstein-Uhlenbeck processes driven by L\'{e}vy motion and their finite and infinite superpositions. We present the general conditions for the $L_q$ convergence of cumulative processes to the limiting processes and investigate their $q$-th order moments and R\'{e}nyi functions, which are nonlinear, hence displaying the multifractality of the processes as constructed. We also establish the corresponding scenarios for the limiting processes, such as log-normal, log-gamma, log-tempered stable or log-normal tempered stable scenarios.