Integer-valued multifractal processes

Abstract

Multifractal scaling has been extensively studied for real-valued stochastic processes, but a systematic integer-valued analogue has remained largely unexplored. In this work, we introduce a multifractal framework for integer-valued processes using the thinning operation, which serves as a natural discrete counterpart to scalar multiplication. Within this framework, we construct integer-valued multifractal processes by time changing compound Poisson processes with nondecreasing multifractal clocks. We derive the scaling laws of their moments, provide explicit examples, and illustrate the results through numerical simulations. This construction integrates multifractal concepts into point process theory, enabling analysis of nonlinear discrete stochastic systems with nontrivial scaling properties.