Scaling properties of correlated random walks

Abstract Many stochastic time series can be modelled by discrete random walks in which a step of random sign but constant length δx is performed after each time interval δt. 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 (∆x,∆t) that a displacement ∆x is observed after a lagtime ∆t is known analytically for arbitrary persistence parameters q. In this short note we show how a CDTRW with parameters [δt, δx, q] can be mapped onto another CDTRW with rescaled parameters [δt/s, δx · g(q, s), q′(q, s)], for arbitrary scaling parameters s, so that both walks have the same displacement distributions P (∆x,∆t) on long time scales. The nonlinear scaling functions g(q, s) and q′(q, s) and derived explicitely. This scaling method can be used to model time series measured at discrete sample intervals δt but actually corresponding to continuum processes with variations occuring on a much shorter time scale δt/s.