Double asymptotic for random walks on hypercubes

TL;DR

This paper investigates the asymptotic behavior of the sum of coordinates of a simple random walk on a hypercube as both the number of steps and the dimension grow large, revealing convergence to different stochastic processes.

Contribution

It establishes a double asymptotic framework for the process, showing how the limiting behavior varies with the ratio of time steps to dimension, unifying several classical limits.

Findings

Convergence to Brownian motion when the ratio tends to zero.

Convergence to Ornstein-Uhlenbeck process at a critical ratio.

Convergence to i.i.d. Gaussian variables when the ratio tends to infinity.

Abstract

We consider the sum of the coordinates of a simple random walk on the K-dimensional hypercube, and prove a double asymptotic of this process, as both the time parameter n and the space parameter K tend to infinity. Depending on the asymptotic ratio of the two parameters, they converge towards either a Brownian motion, an Ornstein-Uhlenbeck process or an i.i.d. collection of Gaussian variables.