Time averages in continuous time random walks

TL;DR

This paper analyzes the behavior of the time averaged squared displacement in continuous time random walks, demonstrating linear growth with the number of steps and showing how fluctuations diminish as the number of steps increases.

Contribution

It provides a rigorous proof that TASD and apparent diffusion constant grow linearly with steps, clarifies fluctuation behavior, and extends some results to correlated steps.

Findings

TASD grows linearly with the number of steps N

Fluctuations in TASD decrease as 1/√N

Non-linear features are suppressed in TASD plots

Abstract

We investigate the time averaged squared displacement (TASD) of continuous time random walks with respect to the number of steps $N$, which the random walker performed during the data acquisition time $T$. We prove that the TASD, and as well the apparent diffusion constant, grow linearly with $N$, provided the steps possess a fourth moment and can not accumulate in small intervals. Consequently, the fluctuations of the latter are dominated by the fluctuations of $N$, and fluctuations of the walker's thermal history are irrelevant. Furthermore, we show that the relative scatter decays as $1/\sqrt{N}$, which suppresses all non-linear features in a plot of the TASD against the lag time. Parts of our arguments also hold for continuous time random walks with correlated steps.