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 $P(\Delta x,\Delta t)$ 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 $\left[  \delta t, \delta x, q \right]$ can be mapped onto another CDTRW with rescaled parameters $\left[  \delta t/s, \delta x\cdot g(q,s), q^{\prime}(q,s) \right]$, 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 $\delta t/s$.

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