Scaling limit of dependent random walk

Jeonghwa Lee

Chaos, Solitons & Fractals, 2026, vol. 208, issue P1

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.

Keywords: Non-Markovian diffusion processes; Weak convergence; Statistical physics; Generalized Bernoulli process; Tempered fractional diffusion equation (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:208:y:2026:i:p1:s0960077926002432

DOI: 10.1016/j.chaos.2026.118102

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