Stroboscopic observation of a random walker

Abstract

The patterns of motion of mobile agents have received recently wide attention in the literature. There is a number of recent studies centered around the motion behavior of many agents ranging from albatrosses to human beings. Special attention has been given to the covered distances statistical distributions. In some cases, due to the lack of accurate data about the motion of the agents it has been necessary to plan very clever experiments to obtain them.  These experiments try to infer the statistical properties of the agents' real motion from the observed positions in consecutive time intervals.  The length of the time intervals are random variables taking their values from a previously known statistical distribution or from a distribution deduced from empirical data. The aim of this work is to demonstrate that for a Gaussian Random Walker it is, in general, impossible to recover the real motion patterns distribution from the stroboscopic observation of the agents unless the length of the intervals between two consecutive measurements are equal. Moreover, it is also shown that the distances distribution strongly depends on the agents' observation time intervals. These claims are sustained by numerical experiments.