Claus Metzner. Scaling properties of correlated random walks

Natural Sciences / Physics / Biophysics

Submitted on: Aug 19, 2012, 19:36:32

Description: 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 $[delta t/s, delta xcdot g(q,s), q^{prime}(q,s)]$, 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.