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 $δx$ 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 $P(Δx,Δt)$ that a displacement $Δx$ is observed after a lagtime $Δt$ 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 $[δt/s, δx\cdot g(q,s), q^{\prime}(q,s)]$, for arbitrary scaling parameters $s$, so that both walks have the same displacement distributions $P(Δx,Δt)$ 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 $δt$ but actually corresponding to continuum processes with variations occuring on a much shorter time scale $δt/s$.

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