Iterated random walk

TL;DR

This paper studies the behavior of iterated random walks in one dimension, deriving their asymptotic density and stability properties using continuum analysis and validating with Monte Carlo simulations.

Contribution

It introduces a continuum limit analysis of iterated random walks and characterizes their asymptotic stable density, a novel approach for this process.

Findings

Asymptotic density is a symmetric exponential function.

Density remains stable under finite modifications of iterations.

Deviation from stationary density decreases exponentially with the number of iterations.

Abstract

The iterated random walk is a random process in which a random walker moves on a one-dimensional random walk which is itself taking place on a one-dimensional random walk, and so on. This process is investigated in the continuum limit using the method of moments. When the number of iterations goes to infinity, a time-independent asymptotic density is obtained. It has a simple symmetric exponential form which is stable against the modification of a finite number of iterations. When n is large, the deviation from the stationary density is exponentially small in n. The continuum results are compared to Monte Carlo data for the discrete iterated random walk.