Comment on: Curious properties of simple random walks (Q2501507)

The authors refer to a paper of \textit{S. I. Ben-Abraham} [J. Stat. Phys. 73, No. 1--2, 441--445 (1993; Zbl 1101.60331], where elementary random walks in one dimension and drift velocities of two random walks relative to each other are examined. The authors of the present note translate Ben-Abraham's results on random walks toward probabilistic terminology and remark that, in case of dimension \(d = 1\), the length of the random walk could be interpreted as the (mean) distance reachable in one unit of time. For higher dimensions (\(d\geq 2\)), this special property of (simple) random walks is not to be expected, due to the incompatibility of a lattice structure with isotropy of space. This is the object of some particular observations of the present note. The authors show that for \(d = 2\) random walks, the interpretation of the length of a random walk as depending on the distance reachable in one unit of time, with the same formula, still holds, provided that some adequate definitions for the probabilistic notions are given. The extension to the dimension \(2d\) seems to be feasible, but a generalization to any natural dimension \(d\) is much less evident (and probable, according to a hint given by the authors).