Limit theorems for multifractal products of geometric stationary 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 Lévy motion and their finite and infinite superpositions. We present the general conditions for the $\mathcal{L}_{q}$ convergence of cumulative processes to the limiting processes and investigate their $q$th order moments and Rényi functions, which are non-linear, 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.