Standard continuous time random walk (CTRW) models are renewal processes in
the sense that at each jump a new, independent pair of jump length and waiting
time are chosen. Globally, anomalous diffusion emerges through action of the
generalized central limit theorem leading to scale-free forms of the jump
length or waiting time distributions. Here we present a modified version of
recently proposed correlated CTRW processes, where we incorporate a power-law
correlated noise on the level of both jump length and waiting time dynamics. We
obtain a very general stochastic model, that encompasses key features of
several paradigmatic models of anomalous diffusion: discontinuous, scale-free
displacements as in Levy flights, scale-free waiting times as in subdiffusive
CTRWs, and the long-range temporal correlations of fractional Brownian motion
(FBM). We derive the exact solutions for the single-time probability density
functions and extract the scaling behaviours. Interestingly, we find that
different combinations of the model parameters lead to indistinguishable shapes
of the emerging probability density functions and identical scaling laws. Our
model will be useful to describe recent experimental single particle tracking
data, that feature a combination of CTRW and FBM properties.Comment: 25 pages, IOP style, 5 figure
