Scaling Limit of Dependent Random Walk
Jeonghwa Lee
TL;DR
This work derives the scaling limit of a dependent random walk driven by a generalized Bernoulli process (GBP), producing a new class of non-Markovian diffusions with stationary increments. Depending on parameter choices, the limiting processes exhibit tempered exponential or Mittag-Leffler-type memory, with marginal densities solving tempered time-fractional (or space-time fractional) diffusion equations. A family of Exponential Mittag-Leffler (EML) processes and their nonstationary counterparts emerge as principal limits, with explicit mgfs and moment structures. Subordinating Lévy processes by these limits yields a broad family of non-Markovian diffusions governed by space-time tempered fractional PDEs and characterized by rich dependence patterns. The results unify scaling limits, explicit distributional forms, and governing PDEs, offering versatile tools for modeling anomalous diffusion in complex systems.
Abstract
Recently, a generalized Bernoulli process (GBP) was developed as a stationary binary sequence that can have long-range dependence. In this paper, we find the scaling limit of a random walk that follows GBP. The result is a new class of non-Markovian diffusion processes. The limiting processes include continuous-time stochastic processes with stationary increments whose correlation decays with an exponential rate, a power law, or an exponentially tempered power law. The limit densities solve a time tempered fractional diffusion equation or time fractional diffusion equation. The second-family of Mittag-Leffler distribution and exponential distribution arise as special cases of the limiting distributions. Subordinated processes are considered as time-changed Lévy processes, and the governing equations and dependence structure of the subordinated processes are discussed.