An explicit calculation of the mean of the perimeter of the convex hull of a plane random walk
Let \((X_ n)_{n\geq 1}\) be i.i.d. random variables in \(R^ 2\), and put \(S_ 0=0\), \(S_ n=\sum^ n_{j=1} X_ j\), \(n\geq 1\). Denote by \(C_ n\) the convex hull of \((S_ k)^ n_{k=0}\) and by \(L_ n\) the perimeter of \(C_ n\). Let \(A_ k\) and \(B_ k\) be the components of \(S_ k\), \(k\geq 0\), and put \(M_ n(\theta)=\max_{0\leq k\leq n} (A_ k\cos \theta+B_ k\sin \theta)\). It is first proved that \(\mathbf{E}(L_ n)= \int_ 0^{2\pi}\text\textbf{E}(M_ n(\theta))d\theta\). Then an example is given where the computation can be made explicitly.
scientific article; zbMATH DE number 179070