Scaling properties of correlated random walks

TL;DR

This paper presents a method to rescale correlated random walks, allowing the modeling of time series sampled at discrete intervals to represent underlying continuous processes with faster dynamics.

Contribution

It introduces explicit nonlinear scaling functions that map correlated discrete random walks onto equivalent walks with different parameters, preserving displacement distributions over long times.

Findings

Derived explicit scaling functions g(q,s) and q'(q,s)

Mapped correlated random walks onto rescaled parameters

Enabled modeling of faster underlying processes from sampled data

Abstract

Many stochastic time series can be modelled by discrete random walks in which a step of random sign but constant length $\delta x$ is performed after each time interval $\delta t$. In correlated discrete time random walks (CDTRWs), the probability $q$ for two successive steps having the same sign is unequal 1/2. The resulting probability distribution $P(\Delta x,\Delta t)$ that a displacement $\Delta x$ is observed after a lagtime $\Delta t$ is known analytically for arbitrary persistence parameters $q$. In this short note we show how a CDTRW with parameters $[\delta t, \delta x, q]$ can be mapped onto another CDTRW with rescaled parameters $[\delta t/s, \delta x\cdot g(q,s), q^{\prime}(q,s)]$, for arbitrary scaling parameters $s$, so that both walks have the same displacement distributions $P(\Delta x,\Delta t)$ on long time scales. The nonlinear scaling functions $g(q,s)$ and $q^{\prime}(q,s)$ and derived explicitely. This scaling method can be used to model time series measured at discrete sample intervals $\delta t$ but actually corresponding to continuum processes with variations occuring on a much shorter time scale $\delta t/s$.