The distribution of the number of crossings of a strip by paths of the simplest random walks and of a Wiener process with drift (Q2882769)

Let \(\xi_1,\dots,\xi_n\) be i.i.d. random variables with \(P(\xi_1=1)=p\), \(P(\xi_1=-1)=q\) and \(P(\xi_1=0)=h\), where \(p+q+h=1\) and \(pq>0\), and set \(S_0=0\), \(S_k=\xi_1+\dots+\xi_k\), \(1\leq k\leq n\). Using purely combinatorial and elementary arguments, the authors first determine the distribution of the number of up-down crossings of the strip \([0,n]\times[-M,N]\) by the random walk \(\{k,S_k\}_{k=0}^n\), where \(M,N\in\mathbb N\). Next, let \(w(t), 0\leq t\leq T\), be a standard Wiener process on the interval \([0,T]\). Then, on account of the Donsker-Prohorov invariance principle, they give the distribution of the number of top-down crossings of the strip \([0,T]\times[-a,b]\) by the process \(w(t)-\beta t, 0\leq t\leq T\), where \(-a<0<b\) and \(\beta\in\mathbb R\).