On the problem of random flights
The author discusses the history (briefly) and mathematics (extensively) of the following. Let \(S_ n=X_ 1+X_ 2+...+X_ n\) where \(X_ i\) is a two dimensional vector of fixed length \(\ell_ i\) but random inclination \(\theta_ i\) to a fixed axis. The \(\theta_ i's\) are independently, identically and uniformly distributed over (-\(\pi\),\(\pi\) ]. The problem of random flights is to find explicitly \(\Pr (| S_ n| \leq r)\), the probability that the vector \(S_ n\) lies within a circle of radius r. The basic solution procedure discussed is that due to \textit{J. C. Kluyver} [Proc. Sect. Sci. Kon. Akad. Wet. Amsterdam 8, 341- 350 (1906)]. Various variants of this random walk problem with independent continuous summands \(X_ i\) are also discussed.
scientific article; zbMATH DE number 3906456
scientific article