Scaling properties of correlated random walks

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 $P(\mathrm{\Delta }x,\mathrm{\Delta }t)$ that a displacement $\mathrm{\Delta }x$ is observed after a lagtime $\mathrm{\Delta }t$ 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 $[\delta t/s, \delta x\cdot g(q,s), q^{\prime}(q,s)]$,forarbitraryscalingparameters$s$, so that both walks have the same displacement distributions $P(\mathrm{\Delta }x,\mathrm{\Delta }t)$ on long time scales. The nonlinear scaling functions $g(q,s)$ and $q^{\mathrm{\prime }}(q,s)$ 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 $\delta t\mathrm{/}s$.